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

Percentage Accurate: 88.0% → 100.0%

Time: 10.9s

Alternatives: 7

Speedup: 1.1×

• 20.3% 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: 88.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

5. Simplified 100.0%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \frac{1 - y}{z}, y\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (/ (- 1.0 y) z) y)))
   (if (<= z -1.55e+39) t_0 (if (<= z 4e-20) (/ (fma (- z x) y x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, ((1.0 - y) / z), y);
	double tmp;
	if (z <= -1.55e+39) {
		tmp = t_0;
	} else if (z <= 4e-20) {
		tmp = fma((z - x), y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, Float64(Float64(1.0 - y) / z), y)
	tmp = 0.0
	if (z <= -1.55e+39)
		tmp = t_0;
	elseif (z <= 4e-20)
		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[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -1.55e+39], t$95$0, If[LessEqual[z, 4e-20], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e39 or 3.99999999999999978e-20 < z

   1. Initial program 80.4%

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

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{z} \cdot \left(1 - y\right)\right)}, y\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1 \cdot \left(1 - y\right)}{\color{blue}{z}}\right), y\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1 - y}{z}\right), y\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(1 - y\right), \color{blue}{z}\right), y\right) \]

      8. --lowering--.f64 99.9%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right), y\right) \]

   7. Applied egg-rr 99.9%

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

   if -1.5500000000000001e39 < z < 3.99999999999999978e-20

   1. Initial program 99.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(z - x\right) + x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(z - x\right) \cdot y + x\right), z\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(z - x\right), y, x\right), z\right) \]

      4. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), y, x\right), z\right) \]

   4. Applied egg-rr 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 3: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -1.32e+163)
     t_0
     (if (<= z 5.5e+148) (/ (fma (- z x) y x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -1.32e+163) {
		tmp = t_0;
	} else if (z <= 5.5e+148) {
		tmp = fma((z - x), y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (z <= -1.32e+163)
		tmp = t_0;
	elseif (z <= 5.5e+148)
		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]), $MachinePrecision]}, If[LessEqual[z, -1.32e+163], t$95$0, If[LessEqual[z, 5.5e+148], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.31999999999999995e163 or 5.5e148 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 68.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. div-inv N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. div-inv N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      10. /-lowering-/.f64 93.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 93.3%

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

   if -1.31999999999999995e163 < z < 5.5e148

   1. Initial program 97.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(z - x\right) + x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(z - x\right) \cdot y + x\right), z\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(z - x\right), y, x\right), z\right) \]

      4. --lowering--.f64 97.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, x\right), y, x\right), z\right) \]

   4. Applied egg-rr 97.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+163}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-5) (/ x z) (if (<= x 8.5e+21) y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-5) {
		tmp = x / z;
	} else if (x <= 8.5e+21) {
		tmp = y;
	} 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 <= (-9.5d-5)) then
        tmp = x / z
    else if (x <= 8.5d+21) then
        tmp = y
    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 <= -9.5e-5) {
		tmp = x / z;
	} else if (x <= 8.5e+21) {
		tmp = y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e-5:
		tmp = x / z
	elif x <= 8.5e+21:
		tmp = y
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-5)
		tmp = Float64(x / z);
	elseif (x <= 8.5e+21)
		tmp = y;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e-5)
		tmp = x / z;
	elseif (x <= 8.5e+21)
		tmp = y;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-5], N[(x / z), $MachinePrecision], If[LessEqual[x, 8.5e+21], y, N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000005e-5 or 8.5e21 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, z\right) \]
   4. Step-by-step derivation

   5. Simplified 63.9%

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

   if -9.5000000000000005e-5 < x < 8.5e21

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.0%

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

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-14) (+ y (/ x z)) (* z (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-14) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / 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 (y <= 2.4d-14) then
        tmp = y + (x / z)
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-14) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-14:
		tmp = y + (x / z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-14)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e-14)
		tmp = y + (x / z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-14], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e-14

   1. Initial program 94.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 83.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. div-inv N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. div-inv N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      10. /-lowering-/.f64 86.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 86.5%

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

   if 2.4e-14 < y

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot z\right)}, z\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot z + 0\right), z\right) \]

      2. accelerator-lowering-fma.f64 28.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(y, z, 0\right), z\right) \]

   5. Simplified 28.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, z, 0\right)}}{z} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      5. /-lowering-/.f64 56.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 56.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in z around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right), z\right) \]
4. Step-by-step derivation

5. Simplified 69.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. div-inv N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. div-inv N/A

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

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

   10. /-lowering-/.f64 75.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

7. Applied egg-rr 75.9%

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

8. Final simplification 75.9%

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

Alternative 7: 41.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 32.3%

   \[\leadsto \color{blue}{y} \]
6. Add Preprocessing

Developer Target 1: 94.1% 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 2024193 
(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))