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

Percentage Accurate: 88.2% → 99.3%

Time: 9.3s

Alternatives: 8

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -1.2e+81)
     t_0
     (if (<= y 6000000000000.0) (/ (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 <= -1.2e+81) {
		tmp = t_0;
	} else if (y <= 6000000000000.0) {
		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 <= -1.2e+81)
		tmp = t_0;
	elseif (y <= 6000000000000.0)
		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, -1.2e+81], t$95$0, If[LessEqual[y, 6000000000000.0], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999995e81 or 6e12 < y

   1. Initial program 69.3%

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

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

      3. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 100.0%

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

   if -1.19999999999999995e81 < y < 6e12

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -9000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.011:\\ \;\;\;\;\frac{x + y \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -9000000000000.0) t_0 (if (<= y 0.011) (/ (+ x (* y z)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -9000000000000.0) {
		tmp = t_0;
	} else if (y <= 0.011) {
		tmp = (x + (y * z)) / 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 <= -9000000000000.0)
		tmp = t_0;
	elseif (y <= 0.011)
		tmp = Float64(Float64(x + Float64(y * z)) / 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, -9000000000000.0], t$95$0, If[LessEqual[y, 0.011], N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e12 or 0.010999999999999999 < y

   1. Initial program 74.4%

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

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 99.5%

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

   if -9e12 < y < 0.010999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -9000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -9000000000000.0) t_0 (if (<= y 0.011) (fma x (/ 1.0 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 <= -9000000000000.0) {
		tmp = t_0;
	} else if (y <= 0.011) {
		tmp = fma(x, (1.0 / 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 <= -9000000000000.0)
		tmp = t_0;
	elseif (y <= 0.011)
		tmp = fma(x, Float64(1.0 / 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, -9000000000000.0], t$95$0, If[LessEqual[y, 0.011], N[(x * N[(1.0 / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e12 or 0.010999999999999999 < y

   1. Initial program 74.4%

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

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 99.5%

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

   if -9e12 < y < 0.010999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.3%

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

Alternative 4: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -9000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (/ (* y x) z))))
   (if (<= y -9000000000000.0) t_0 (if (<= y 0.011) (fma x (/ 1.0 z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((y * x) / z);
	double tmp;
	if (y <= -9000000000000.0) {
		tmp = t_0;
	} else if (y <= 0.011) {
		tmp = fma(x, (1.0 / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(y * x) / z))
	tmp = 0.0
	if (y <= -9000000000000.0)
		tmp = t_0;
	elseif (y <= 0.011)
		tmp = fma(x, Float64(1.0 / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9000000000000.0], t$95$0, If[LessEqual[y, 0.011], N[(x * N[(1.0 / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e12 or 0.010999999999999999 < y

   1. Initial program 74.4%

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

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   if -9e12 < y < 0.010999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9000000000000:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{y \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.35e+31) (/ x z) (if (<= x 6.4e-70) (/ (* y z) z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35e+31) {
		tmp = x / z;
	} else if (x <= 6.4e-70) {
		tmp = (y * z) / z;
	} else {
		tmp = x / z;
	}
	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 (x <= (-1.35d+31)) then
        tmp = x / z
    else if (x <= 6.4d-70) then
        tmp = (y * z) / z
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35e+31) {
		tmp = x / z;
	} else if (x <= 6.4e-70) {
		tmp = (y * z) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.35e+31:
		tmp = x / z
	elif x <= 6.4e-70:
		tmp = (y * z) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.35e+31)
		tmp = Float64(x / z);
	elseif (x <= 6.4e-70)
		tmp = Float64(Float64(y * z) / z);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.35e+31)
		tmp = x / z;
	elseif (x <= 6.4e-70)
		tmp = (y * z) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.35e+31], N[(x / z), $MachinePrecision], If[LessEqual[x, 6.4e-70], N[(N[(y * z), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999993e31 or 6.3999999999999995e-70 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if -1.34999999999999993e31 < x < 6.3999999999999995e-70

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

Alternative 6: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 43000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 - y}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 43000000000000.0) (fma x (/ (- 1.0 y) z) y) (fma (- y) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 43000000000000.0) {
		tmp = fma(x, ((1.0 - y) / z), y);
	} else {
		tmp = fma(-y, (x / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 43000000000000.0)
		tmp = fma(x, Float64(Float64(1.0 - y) / z), y);
	else
		tmp = fma(Float64(-y), Float64(x / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 43000000000000.0], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e13

   1. Initial program 91.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.7%

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

   if 4.3e13 < y

   1. Initial program 72.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 100.0%

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

Alternative 7: 77.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 95.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 82.9%

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

Alternative 8: 39.6% 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 86.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 38.0

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

5. Applied rewrites 38.0%

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

Developer Target 1: 93.8% 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 2024232 
(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))