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

Percentage Accurate: 88.0% → 98.6%

Time: 6.0s

Alternatives: 8

Speedup: 0.5×

• 20.6% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-73} \lor \neg \left(z \leq 2.1 \cdot 10^{-185}\right):\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.5e-73) (not (<= z 2.1e-185)))
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5e-73) || !(z <= 2.1e-185)) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else {
		tmp = (x + (y * (z - 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 ((z <= (-7.5d-73)) .or. (.not. (z <= 2.1d-185))) then
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5e-73) || !(z <= 2.1e-185)) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.5e-73) or not (z <= 2.1e-185):
		tmp = (x / z) + (y * (1.0 - (x / z)))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.5e-73) || !(z <= 2.1e-185))
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.5e-73) || ~((z <= 2.1e-185)))
		tmp = (x / z) + (y * (1.0 - (x / z)));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.5e-73], N[Not[LessEqual[z, 2.1e-185]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e-73 or 2.1e-185 < z

   1. Initial program 81.1%

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

   3. Taylor expanded in y around 0 100.0%

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

   if -7.5e-73 < z < 2.1e-185

   1. Initial program 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-73} \lor \neg \left(z \leq 2.1 \cdot 10^{-185}\right):\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 20000000000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+37)
   (* y (- 1.0 (/ x z)))
   (if (<= y 20000000000000.0) (/ (+ x (* y (- z x))) z) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+37) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 20000000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y * (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 (y <= (-2.2d+37)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 20000000000000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+37) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 20000000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e+37:
		tmp = y * (1.0 - (x / z))
	elif y <= 20000000000000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+37)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 20000000000000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e+37)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 20000000000000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+37], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 20000000000000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000001e37

   1. Initial program 65.7%

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

   3. Taylor expanded in y around inf 65.7%

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

      1. associate-/l* 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. *-inverses 100.0%

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

      5. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -2.2000000000000001e37 < y < 2e13

   1. Initial program 99.9%

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

   if 2e13 < y

   1. Initial program 74.3%

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

   3. Taylor expanded in y around inf 74.3%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. distribute-neg-frac2 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 88.3%

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

      3. add-sqr-sqrt 45.7%

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

      4. sqrt-unprod 62.2%

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

      5. sqr-neg 62.2%

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

      6. sqrt-unprod 21.4%

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

      7. add-sqr-sqrt 41.3%

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

      8. associate-*r/ 46.8%

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

      9. *-commutative 46.8%

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

      10. cancel-sign-sub 46.8%

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

      11. distribute-frac-neg2 46.8%

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

      12. *-commutative 46.8%

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

      13. associate-*r/ 41.3%

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

      14. add-sqr-sqrt 19.9%

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

      15. sqrt-unprod 57.9%

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

      16. sqr-neg 57.9%

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

      17. sqrt-unprod 42.6%

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

      18. add-sqr-sqrt 88.3%

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

      19. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+32} \lor \neg \left(y \leq 0.0155\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.2e+32) (not (<= y 0.0155)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+32) || !(y <= 0.0155)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (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 ((y <= (-5.2d+32)) .or. (.not. (y <= 0.0155d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+32) || !(y <= 0.0155)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.2e+32) or not (y <= 0.0155):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.2e+32) || !(y <= 0.0155))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.2e+32) || ~((y <= 0.0155)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.2e+32], N[Not[LessEqual[y, 0.0155]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000004e32 or 0.0155 < y

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf 71.3%

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

      1. associate-/l* 99.5%

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

      2. div-sub 99.6%

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

      3. sub-neg 99.6%

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

      4. *-inverses 99.6%

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

      5. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -5.2000000000000004e32 < y < 0.0155

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 91.2%

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

   4. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+32} \lor \neg \left(y \leq 0.0155\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 0.0155:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+32)
   (* y (- 1.0 (/ x z)))
   (if (<= y 0.0155) (+ y (/ x z)) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+32) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 0.0155) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (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 (y <= (-5.2d+32)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 0.0155d0) then
        tmp = y + (x / z)
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+32) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 0.0155) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e+32:
		tmp = y * (1.0 - (x / z))
	elif y <= 0.0155:
		tmp = y + (x / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e+32)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 0.0155)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.2e+32)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 0.0155)
		tmp = y + (x / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e+32], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0155], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2000000000000004e32

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf 67.0%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

   if -5.2000000000000004e32 < y < 0.0155

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 91.2%

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

   4. Taylor expanded in x around 0 99.0%

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

   if 0.0155 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in y around inf 74.4%

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

      1. associate-/l* 99.3%

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

      2. div-sub 99.3%

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

      3. sub-neg 99.3%

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

      4. *-inverses 99.3%

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

      5. sub-neg 99.3%

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

   5. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-rgt-in 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. distribute-neg-frac2 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r/ 88.0%

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

      3. add-sqr-sqrt 45.2%

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

      4. sqrt-unprod 62.7%

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

      5. sqr-neg 62.7%

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

      6. sqrt-unprod 22.3%

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

      7. add-sqr-sqrt 41.6%

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

      8. associate-*r/ 46.9%

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

      9. *-commutative 46.9%

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

      10. cancel-sign-sub 46.9%

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

      11. distribute-frac-neg2 46.9%

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

      12. *-commutative 46.9%

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

      13. associate-*r/ 41.6%

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

      14. add-sqr-sqrt 19.3%

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

      15. sqrt-unprod 57.7%

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

      16. sqr-neg 57.7%

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

      17. sqrt-unprod 42.8%

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

      18. add-sqr-sqrt 88.0%

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

      19. associate-*r/ 99.3%

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

   9. Applied egg-rr 99.3%

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

Alternative 5: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e-14) y (if (<= y 4.2e-18) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e-14) {
		tmp = y;
	} else if (y <= 4.2e-18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.5d-14)) then
        tmp = y
    else if (y <= 4.2d-18) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e-14) {
		tmp = y;
	} else if (y <= 4.2e-18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.5e-14:
		tmp = y
	elif y <= 4.2e-18:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e-14)
		tmp = y;
	elseif (y <= 4.2e-18)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.5e-14)
		tmp = y;
	elseif (y <= 4.2e-18)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.5e-14], y, If[LessEqual[y, 4.2e-18], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999991e-14 or 4.19999999999999999e-18 < y

   1. Initial program 74.2%

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

   3. Taylor expanded in x around 0 52.6%

      \[\leadsto \color{blue}{y} \]

   if -5.49999999999999991e-14 < y < 4.19999999999999999e-18

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.8%

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

Alternative 6: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0155:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0155) (+ y (/ x z)) (- y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0155) {
		tmp = y + (x / z);
	} else {
		tmp = y - (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 (y <= 0.0155d0) then
        tmp = y + (x / z)
    else
        tmp = y - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0155) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0155:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0155)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0155)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0155], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0155

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0 93.6%

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

   4. Taylor expanded in x around 0 90.2%

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

   if 0.0155 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in y around 0 86.6%

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

   4. Taylor expanded in x around 0 47.4%

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

      1. add-sqr-sqrt 24.5%

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

      2. sqrt-unprod 54.1%

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

      3. sqr-neg 54.1%

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

      4. sqrt-unprod 28.0%

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

      5. add-sqr-sqrt 61.9%

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

      6. distribute-frac-neg2 61.9%

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

      7. sub-neg 61.9%

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

   6. Applied egg-rr 61.9%

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

Alternative 7: 78.3% accurate, 1.8× 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 86.8%

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

3. Taylor expanded in y around 0 91.8%

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

4. Taylor expanded in x around 0 78.8%

   \[\leadsto \color{blue}{y} + \frac{x}{z} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 9.0× 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 86.8%

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

3. Taylor expanded in x around 0 41.1%

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

Developer Target 1: 93.8% accurate, 0.8× 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 2024170 
(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))