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

Percentage Accurate: 88.4% → 98.2%

Time: 8.2s

Alternatives: 11

Speedup: 0.6×

• 20.4% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+144} \lor \neg \left(z \leq 10^{+61}\right):\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.8e+144) (not (<= z 1e+61)))
   (+ y (* x (/ (- 1.0 y) z)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.8e+144) || !(z <= 1e+61)) {
		tmp = y + (x * ((1.0 - y) / 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 <= (-6.8d+144)) .or. (.not. (z <= 1d+61))) then
        tmp = y + (x * ((1.0d0 - y) / 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 <= -6.8e+144) || !(z <= 1e+61)) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.8e+144) or not (z <= 1e+61):
		tmp = y + (x * ((1.0 - y) / z))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.8e+144) || !(z <= 1e+61))
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / 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 <= -6.8e+144) || ~((z <= 1e+61)))
		tmp = y + (x * ((1.0 - y) / z));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.8e+144], N[Not[LessEqual[z, 1e+61]], $MachinePrecision]], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $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 < -6.7999999999999998e144 or 9.99999999999999949e60 < z

   1. Initial program 60.2%

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

   3. Taylor expanded in x around inf 91.3%

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

   4. Taylor expanded in y around 0 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. *-rgt-identity 91.3%

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

      4. associate-*r/ 91.3%

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

      5. associate-/l* 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sub-neg 99.9%

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

      9. div-sub 99.9%

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

   6. Simplified 99.9%

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

   if -6.7999999999999998e144 < z < 9.99999999999999949e60

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+144} \lor \neg \left(z \leq 10^{+61}\right):\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+48}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+20}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e+48)
   (- y (* y (/ x z)))
   (if (<= y 5e+20) (+ y (* x (/ (- 1.0 y) z))) (* y (/ (- z x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+48) {
		tmp = y - (y * (x / z));
	} else if (y <= 5e+20) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = 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 (y <= (-5d+48)) then
        tmp = y - (y * (x / z))
    else if (y <= 5d+20) then
        tmp = y + (x * ((1.0d0 - y) / z))
    else
        tmp = y * ((z - x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+48) {
		tmp = y - (y * (x / z));
	} else if (y <= 5e+20) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = y * ((z - x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e+48:
		tmp = y - (y * (x / z))
	elif y <= 5e+20:
		tmp = y + (x * ((1.0 - y) / z))
	else:
		tmp = y * ((z - x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e+48)
		tmp = Float64(y - Float64(y * Float64(x / z)));
	elseif (y <= 5e+20)
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / z)));
	else
		tmp = Float64(y * Float64(Float64(z - x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e+48)
		tmp = y - (y * (x / z));
	elseif (y <= 5e+20)
		tmp = y + (x * ((1.0 - y) / z));
	else
		tmp = y * ((z - x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e+48], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+20], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999973e48

   1. Initial program 72.6%

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

   3. Taylor expanded in x around inf 95.2%

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

   4. Taylor expanded in y around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. distribute-rgt-neg-in 95.2%

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

   6. Simplified 95.2%

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

      1. div-inv 95.2%

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

      2. add-sqr-sqrt 95.2%

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

      3. sqrt-unprod 57.9%

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

      4. sqr-neg 57.9%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 61.6%

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

      7. remove-double-neg 61.6%

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

      8. distribute-rgt-neg-out 61.6%

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

      9. cancel-sign-sub-inv 61.6%

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

      10. div-inv 61.6%

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

      11. *-commutative 61.6%

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

      12. associate-/l* 64.6%

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

      13. add-sqr-sqrt 64.6%

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

      14. sqrt-unprod 25.4%

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

      15. sqr-neg 25.4%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -4.99999999999999973e48 < y < 5e20

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf 99.3%

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

   4. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. associate-*r/ 99.2%

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

      5. associate-/l* 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. sub-neg 99.8%

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

      9. div-sub 99.8%

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

   6. Simplified 99.8%

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

   if 5e20 < 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 72.5%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+48}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+20}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+64} \lor \neg \left(x \leq 6 \cdot 10^{+39}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.2e+64) (not (<= x 6e+39)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.2e+64) || !(x <= 6e+39)) {
		tmp = x * ((1.0 - y) / 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 ((x <= (-8.2d+64)) .or. (.not. (x <= 6d+39))) then
        tmp = x * ((1.0d0 - y) / 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 ((x <= -8.2e+64) || !(x <= 6e+39)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.2e+64) or not (x <= 6e+39):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.2e+64) || !(x <= 6e+39))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.2e+64) || ~((x <= 6e+39)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.2e+64], N[Not[LessEqual[x, 6e+39]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.19999999999999956e64 or 5.9999999999999999e39 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

   5. Simplified 90.1%

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

   if -8.19999999999999956e64 < x < 5.9999999999999999e39

   1. Initial program 84.3%

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

   3. Taylor expanded in z around inf 74.7%

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

   4. Taylor expanded in x around 0 90.3%

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

      1. +-commutative 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+64} \lor \neg \left(x \leq 6 \cdot 10^{+39}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1350000000 \lor \neg \left(y \leq 0.001\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1350000000.0) (not (<= y 0.001)))
   (* y (/ (- z x) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1350000000.0) || !(y <= 0.001)) {
		tmp = y * ((z - 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 <= (-1350000000.0d0)) .or. (.not. (y <= 0.001d0))) then
        tmp = y * ((z - 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 <= -1350000000.0) || !(y <= 0.001)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1350000000.0) or not (y <= 0.001):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1350000000.0) || !(y <= 0.001))
		tmp = Float64(y * Float64(Float64(z - 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 <= -1350000000.0) || ~((y <= 0.001)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1350000000.0], N[Not[LessEqual[y, 0.001]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e9 or 1e-3 < y

   1. Initial program 74.0%

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

   3. Taylor expanded in y around inf 73.3%

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

      1. associate-/l* 99.2%

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

   5. Simplified 99.2%

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

   if -1.35e9 < y < 1e-3

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf 97.5%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1350000000 \lor \neg \left(y \leq 0.001\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1350000000.0)
   (- y (* y (/ x z)))
   (if (<= y 0.001) (+ y (/ x z)) (* y (/ (- z x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1350000000.0) {
		tmp = y - (y * (x / z));
	} else if (y <= 0.001) {
		tmp = y + (x / z);
	} else {
		tmp = 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 (y <= (-1350000000.0d0)) then
        tmp = y - (y * (x / z))
    else if (y <= 0.001d0) then
        tmp = y + (x / z)
    else
        tmp = y * ((z - x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1350000000.0) {
		tmp = y - (y * (x / z));
	} else if (y <= 0.001) {
		tmp = y + (x / z);
	} else {
		tmp = y * ((z - x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1350000000.0:
		tmp = y - (y * (x / z))
	elif y <= 0.001:
		tmp = y + (x / z)
	else:
		tmp = y * ((z - x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1350000000.0)
		tmp = Float64(y - Float64(y * Float64(x / z)));
	elseif (y <= 0.001)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(Float64(z - x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1350000000.0)
		tmp = y - (y * (x / z));
	elseif (y <= 0.001)
		tmp = y + (x / z);
	else
		tmp = y * ((z - x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1350000000.0], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.001], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e9

   1. Initial program 73.3%

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

   3. Taylor expanded in x around inf 94.5%

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

   4. Taylor expanded in y around inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-in 94.2%

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

   6. Simplified 94.2%

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

      1. div-inv 94.2%

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

      2. add-sqr-sqrt 94.2%

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

      3. sqrt-unprod 61.6%

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

      4. sqr-neg 61.6%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 58.0%

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

      7. remove-double-neg 58.0%

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

      8. distribute-rgt-neg-out 58.0%

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

      9. cancel-sign-sub-inv 58.0%

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

      10. div-inv 58.0%

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

      11. *-commutative 58.0%

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

      12. associate-/l* 60.7%

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

      13. add-sqr-sqrt 60.7%

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

      14. sqrt-unprod 26.4%

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

      15. sqr-neg 26.4%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 99.7%

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

   8. Applied egg-rr 99.7%

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

   if -1.35e9 < y < 1e-3

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf 97.5%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

   6. Simplified 98.3%

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

   if 1e-3 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in y around inf 73.6%

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

      1. associate-/l* 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{z} - y\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;-\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+228)
   (- (/ x z) y)
   (if (<= x -1.65e+142) (- (* (/ y z) x)) (+ y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+228) {
		tmp = (x / z) - y;
	} else if (x <= -1.65e+142) {
		tmp = -((y / z) * x);
	} 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 (x <= (-9d+228)) then
        tmp = (x / z) - y
    else if (x <= (-1.65d+142)) then
        tmp = -((y / z) * x)
    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 (x <= -9e+228) {
		tmp = (x / z) - y;
	} else if (x <= -1.65e+142) {
		tmp = -((y / z) * x);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e+228:
		tmp = (x / z) - y
	elif x <= -1.65e+142:
		tmp = -((y / z) * x)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+228)
		tmp = Float64(Float64(x / z) - y);
	elseif (x <= -1.65e+142)
		tmp = Float64(-Float64(Float64(y / z) * x));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e+228)
		tmp = (x / z) - y;
	elseif (x <= -1.65e+142)
		tmp = -((y / z) * x);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+228], N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, -1.65e+142], (-N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]), N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999966e228

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 87.6%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   6. Simplified 87.6%

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

      1. add-sqr-sqrt 38.5%

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

      2. sqrt-unprod 95.3%

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

      3. sqr-neg 95.3%

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

      4. sqrt-unprod 49.6%

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

      5. add-sqr-sqrt 88.1%

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

      6. sub-neg 88.1%

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

   8. Applied egg-rr 88.1%

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

   if -8.99999999999999966e228 < x < -1.6500000000000001e142

   1. Initial program 87.3%

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

   3. Taylor expanded in x around inf 87.3%

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

      1. associate-/l* 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-neg-frac2 80.6%

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

   8. Simplified 80.6%

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

   if -1.6500000000000001e142 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around inf 70.7%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{z} - y\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;-\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{z} - y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e+232)
   (- (/ x z) y)
   (if (<= x -4.6e+139) (* y (/ x (- z))) (+ y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e+232) {
		tmp = (x / z) - y;
	} else if (x <= -4.6e+139) {
		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 (x <= (-1.95d+232)) then
        tmp = (x / z) - y
    else if (x <= (-4.6d+139)) 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 (x <= -1.95e+232) {
		tmp = (x / z) - y;
	} else if (x <= -4.6e+139) {
		tmp = y * (x / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e+232:
		tmp = (x / z) - y
	elif x <= -4.6e+139:
		tmp = y * (x / -z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e+232)
		tmp = Float64(Float64(x / z) - y);
	elseif (x <= -4.6e+139)
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.95e+232)
		tmp = (x / z) - y;
	elseif (x <= -4.6e+139)
		tmp = y * (x / -z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e+232], N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, -4.6e+139], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9499999999999999e232

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 87.6%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. +-commutative 87.6%

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

   6. Simplified 87.6%

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

      1. add-sqr-sqrt 38.5%

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

      2. sqrt-unprod 95.3%

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

      3. sqr-neg 95.3%

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

      4. sqrt-unprod 49.6%

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

      5. add-sqr-sqrt 88.1%

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

      6. sub-neg 88.1%

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

   8. Applied egg-rr 88.1%

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

   if -1.9499999999999999e232 < x < -4.6e139

   1. Initial program 87.3%

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

   3. Taylor expanded in y around inf 68.3%

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

      1. associate-/l* 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in z around 0 68.2%

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

      1. mul-1-neg 68.2%

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

      2. associate-*l/ 80.7%

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

      3. *-commutative 80.7%

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

      4. distribute-rgt-neg-in 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   8. Simplified 80.7%

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

   if -4.6e139 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in z around inf 70.7%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{z} - y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+158}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.2e+158) (+ y (/ (* x (- 1.0 y)) z)) (+ y (* x (/ (- 1.0 y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.2e+158) {
		tmp = y + ((x * (1.0 - y)) / z);
	} else {
		tmp = y + (x * ((1.0 - 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 (z <= 6.2d+158) then
        tmp = y + ((x * (1.0d0 - y)) / z)
    else
        tmp = y + (x * ((1.0d0 - y) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.2e+158) {
		tmp = y + ((x * (1.0 - y)) / z);
	} else {
		tmp = y + (x * ((1.0 - y) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.2e+158:
		tmp = y + ((x * (1.0 - y)) / z)
	else:
		tmp = y + (x * ((1.0 - y) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.2e+158)
		tmp = Float64(y + Float64(Float64(x * Float64(1.0 - y)) / z));
	else
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 6.2e+158)
		tmp = y + ((x * (1.0 - y)) / z);
	else
		tmp = y + (x * ((1.0 - y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.2e+158], N[(y + N[(N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.2000000000000004e158

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf 99.1%

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

   if 6.2000000000000004e158 < z

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf 86.9%

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

   4. Taylor expanded in y around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. *-rgt-identity 86.9%

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

      4. associate-*r/ 86.8%

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

      5. associate-/l* 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sub-neg 99.9%

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

      9. div-sub 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+158}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+20} \lor \neg \left(x \leq 5.6 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e+20) (not (<= x 5.6e+39))) (/ x z) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e+20) || !(x <= 5.6e+39)) {
		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 ((x <= (-5.8d+20)) .or. (.not. (x <= 5.6d+39))) 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 ((x <= -5.8e+20) || !(x <= 5.6e+39)) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.8e+20) or not (x <= 5.6e+39):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.8e+20) || !(x <= 5.6e+39))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.8e+20) || ~((x <= 5.6e+39)))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e+20], N[Not[LessEqual[x, 5.6e+39]], $MachinePrecision]], N[(x / z), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e20 or 5.60000000000000003e39 < x

   1. Initial program 89.9%

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

   3. Taylor expanded in y around 0 55.0%

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

   if -5.8e20 < x < 5.60000000000000003e39

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 73.3%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+20} \lor \neg \left(x \leq 5.6 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% 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 86.3%

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

3. Taylor expanded in z around inf 68.8%

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

4. Taylor expanded in x around 0 80.3%

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

   1. +-commutative 80.3%

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

6. Simplified 80.3%

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

7. Final simplification 80.3%

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

Alternative 11: 40.6% 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 86.3%

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

3. Taylor expanded in x around 0 48.3%

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

4. Final simplification 48.3%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 93.9% 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 2024044 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))