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

Percentage Accurate: 87.7% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.1×

• 20.9% 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: 87.7% 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, 1.1× 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 88.5%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6400:\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{elif}\;y \leq 0.048:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6400.0)
   (* (/ (- z x) z) y)
   (if (<= y 0.048) (fma (/ x z) 1.0 y) (fma (/ x z) (- y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6400.0) {
		tmp = ((z - x) / z) * y;
	} else if (y <= 0.048) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = fma((x / z), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6400.0)
		tmp = Float64(Float64(Float64(z - x) / z) * y);
	elseif (y <= 0.048)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = fma(Float64(x / z), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6400.0], N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 0.048], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6400

   1. Initial program 72.9%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -6400 < y < 0.048000000000000001

   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-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ 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 98.9%

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

   if 0.048000000000000001 < 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

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.9%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z - x}{z} \cdot y\\ \mathbf{if}\;y \leq -6400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.048:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- z x) z) y)))
   (if (<= y -6400.0) t_0 (if (<= y 0.048) (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 <= -6400.0) {
		tmp = t_0;
	} else if (y <= 0.048) {
		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 <= -6400.0)
		tmp = t_0;
	elseif (y <= 0.048)
		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, -6400.0], t$95$0, If[LessEqual[y, 0.048], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6400 or 0.048000000000000001 < y

   1. Initial program 76.0%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -6400 < y < 0.048000000000000001

   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-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ 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 98.9%

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

Alternative 4: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 1.0 y)))
   (if (<= z -1.6e-105) t_0 (if (<= z 2.2e-7) (* (- 1.0 y) (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 1.0, y);
	double tmp;
	if (z <= -1.6e-105) {
		tmp = t_0;
	} else if (z <= 2.2e-7) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 1.0, y)
	tmp = 0.0
	if (z <= -1.6e-105)
		tmp = t_0;
	elseif (z <= 2.2e-7)
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]}, If[LessEqual[z, -1.6e-105], t$95$0, If[LessEqual[z, 2.2e-7], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.59999999999999991e-105 or 2.2000000000000001e-7 < z

   1. Initial program 81.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ 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 90.3%

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

   if -1.59999999999999991e-105 < z < 2.2000000000000001e-7

   1. Initial program 99.9%

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

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot y}{z}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z y) z)))
   (if (<= y -4.2e-20) t_0 (if (<= y 7.5e-6) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) / z;
	double tmp;
	if (y <= -4.2e-20) {
		tmp = t_0;
	} else if (y <= 7.5e-6) {
		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 <= (-4.2d-20)) then
        tmp = t_0
    else if (y <= 7.5d-6) 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 <= -4.2e-20) {
		tmp = t_0;
	} else if (y <= 7.5e-6) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * y) / z
	tmp = 0
	if y <= -4.2e-20:
		tmp = t_0
	elif y <= 7.5e-6:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * y) / z)
	tmp = 0.0
	if (y <= -4.2e-20)
		tmp = t_0;
	elseif (y <= 7.5e-6)
		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 <= -4.2e-20)
		tmp = t_0;
	elseif (y <= 7.5e-6)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -4.2e-20], t$95$0, If[LessEqual[y, 7.5e-6], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999998e-20 or 7.50000000000000019e-6 < y

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   if -4.1999999999999998e-20 < y < 7.50000000000000019e-6

   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 76.2

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

   5. Applied rewrites 76.2%

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

Alternative 6: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.55e+213) (fma (/ x z) 1.0 y) (/ (* (- y) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.55e+213) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = (-y * x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.55e+213)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = Float64(Float64(Float64(-y) * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.55e+213], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[(N[((-y) * x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5500000000000001e213

   1. Initial program 89.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ 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 84.9%

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

   if 2.5500000000000001e213 < y

   1. Initial program 79.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.9%

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

Alternative 7: 78.0% 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 88.5%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ 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 81.7%

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

Alternative 8: 39.7% 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 88.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 42.3

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

5. Applied rewrites 42.3%

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

Developer Target 1: 93.6% 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 2024298 
(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))