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

Percentage Accurate: 88.4% → 99.8%

Time: 7.7s

Alternatives: 8

Speedup: 0.7×

• 20.5% 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: 88.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -1.58 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 59000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -1.58e+26)
     t_0
     (if (<= y 59000000000000.0) (/ (fma (- z x) y x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -1.58e+26) {
		tmp = t_0;
	} else if (y <= 59000000000000.0) {
		tmp = fma((z - x), y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-y), Float64(x / z), y)
	tmp = 0.0
	if (y <= -1.58e+26)
		tmp = t_0;
	elseif (y <= 59000000000000.0)
		tmp = Float64(fma(Float64(z - x), y, x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -1.58e+26], t$95$0, If[LessEqual[y, 59000000000000.0], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.57999999999999994e26 or 5.9e13 < y

   1. Initial program 70.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 70.8

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in y around inf

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

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.57999999999999994e26 < y < 5.9e13

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -2.6e+60) t_0 (if (<= y 5e+19) (fma (/ (- 1.0 y) z) x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -2.6e+60) {
		tmp = t_0;
	} else if (y <= 5e+19) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-y), Float64(x / z), y)
	tmp = 0.0
	if (y <= -2.6e+60)
		tmp = t_0;
	elseif (y <= 5e+19)
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -2.6e+60], t$95$0, If[LessEqual[y, 5e+19], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000008e60 or 5e19 < y

   1. Initial program 69.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in y around inf

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

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -2.60000000000000008e60 < y < 5e19

   1. Initial program 99.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.7%

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

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -58000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -58000000.0) t_0 (if (<= y 0.5) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -58000000.0) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-y), Float64(x / z), y)
	tmp = 0.0
	if (y <= -58000000.0)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -58000000.0], t$95$0, If[LessEqual[y, 0.5], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8e7 or 0.5 < y

   1. Initial program 72.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.6

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around inf

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

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -5.8e7 < y < 0.5

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.7%

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

Alternative 4: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - y}{z} \cdot x\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- 1.0 y) z) x)))
   (if (<= x -2.4e+117) t_0 (if (<= x 1.02e+189) (fma 1.0 (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - y) / z) * x;
	double tmp;
	if (x <= -2.4e+117) {
		tmp = t_0;
	} else if (x <= 1.02e+189) {
		tmp = fma(1.0, (x / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - y) / z) * x)
	tmp = 0.0
	if (x <= -2.4e+117)
		tmp = t_0;
	elseif (x <= 1.02e+189)
		tmp = fma(1.0, Float64(x / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.4e+117], t$95$0, If[LessEqual[x, 1.02e+189], N[(1.0 * N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3999999999999999e117 or 1.02e189 < x

   1. Initial program 89.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   if -2.3999999999999999e117 < x < 1.02e189

   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. Taylor expanded in y around 0

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

   8. Applied rewrites 87.7%

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

   10. Applied rewrites 87.9%

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

Alternative 5: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot y}{z}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, y, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z y) z)))
   (if (<= y -3.2e-18) t_0 (if (<= y 4.8e-35) (/ x (fma z y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) / z;
	double tmp;
	if (y <= -3.2e-18) {
		tmp = t_0;
	} else if (y <= 4.8e-35) {
		tmp = x / fma(z, y, 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 <= -3.2e-18)
		tmp = t_0;
	elseif (y <= 4.8e-35)
		tmp = Float64(x / fma(z, y, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -3.2e-18], t$95$0, If[LessEqual[y, 4.8e-35], N[(x / N[(z * y + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999999e-18 or 4.8000000000000003e-35 < y

   1. Initial program 75.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if -3.1999999999999999e-18 < y < 4.8000000000000003e-35

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 4.8%

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 77.6%

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

Alternative 6: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot y}{z}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z y) z)))
   (if (<= y -3.2e-18) t_0 (if (<= y 4.8e-35) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) / z;
	double tmp;
	if (y <= -3.2e-18) {
		tmp = t_0;
	} else if (y <= 4.8e-35) {
		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 <= (-3.2d-18)) then
        tmp = t_0
    else if (y <= 4.8d-35) 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 <= -3.2e-18) {
		tmp = t_0;
	} else if (y <= 4.8e-35) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * y) / z
	tmp = 0
	if y <= -3.2e-18:
		tmp = t_0
	elif y <= 4.8e-35:
		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 <= -3.2e-18)
		tmp = t_0;
	elseif (y <= 4.8e-35)
		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 <= -3.2e-18)
		tmp = t_0;
	elseif (y <= 4.8e-35)
		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, -3.2e-18], t$95$0, If[LessEqual[y, 4.8e-35], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999999e-18 or 4.8000000000000003e-35 < y

   1. Initial program 75.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if -3.1999999999999999e-18 < y < 4.8000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

Alternative 7: 78.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.2%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 95.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 82.8%

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

10. Applied rewrites 83.0%

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

Alternative 8: 39.5% 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 87.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 42.9

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

5. Applied rewrites 42.9%

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

Developer Target 1: 94.2% 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 2024270 
(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))