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

Percentage Accurate: 88.1% → 99.6%

Time: 9.4s

Alternatives: 12

Speedup: 0.6×

• 21.0% 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: 88.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-223}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-223)
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (if (<= y 0.00078) (/ (+ x (* y (- z x))) z) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-223) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 0.00078) {
		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 <= (-4d-223)) then
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    else if (y <= 0.00078d0) 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 <= -4e-223) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 0.00078) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-223:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	elif y <= 0.00078:
		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 <= -4e-223)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	elseif (y <= 0.00078)
		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 <= -4e-223)
		tmp = (x / z) + (y * (1.0 - (x / z)));
	elseif (y <= 0.00078)
		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, -4e-223], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00078], 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 < -3.9999999999999999e-223

   1. Initial program 80.6%

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

   3. Taylor expanded in y around 0 99.9%

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

   if -3.9999999999999999e-223 < y < 7.79999999999999986e-4

   1. Initial program 100.0%

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

   if 7.79999999999999986e-4 < y

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0 95.7%

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

   4. Taylor expanded in y around inf 91.8%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

      3. distribute-rgt-neg-in 95.7%

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

      4. distribute-frac-neg2 95.7%

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

   6. Simplified 95.7%

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

      1. *-commutative 95.7%

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

      2. add-sqr-sqrt 46.3%

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

      3. sqrt-unprod 65.0%

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

      4. sqr-neg 65.0%

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

      5. sqrt-unprod 25.2%

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

      6. add-sqr-sqrt 45.7%

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

      7. cancel-sign-sub 45.7%

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

      8. distribute-frac-neg2 45.7%

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

      9. associate-*l/ 43.3%

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

      10. *-commutative 43.3%

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

      11. add-sqr-sqrt 20.4%

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

      12. sqrt-unprod 62.8%

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

      13. sqr-neg 62.8%

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

      14. sqrt-unprod 46.7%

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

      15. add-sqr-sqrt 91.8%

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

      16. associate-*l/ 100.0%

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

      17. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-223}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+275}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e+26)
   (+ y (/ x z))
   (if (<= y 3.7e+112)
     (* y (/ (- x) z))
     (if (<= y 8.2e+275) (* z (/ y z)) (* x (- (/ y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+26) {
		tmp = y + (x / z);
	} else if (y <= 3.7e+112) {
		tmp = y * (-x / z);
	} else if (y <= 8.2e+275) {
		tmp = z * (y / z);
	} else {
		tmp = x * -(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 <= 1.7d+26) then
        tmp = y + (x / z)
    else if (y <= 3.7d+112) then
        tmp = y * (-x / z)
    else if (y <= 8.2d+275) then
        tmp = z * (y / z)
    else
        tmp = x * -(y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+26) {
		tmp = y + (x / z);
	} else if (y <= 3.7e+112) {
		tmp = y * (-x / z);
	} else if (y <= 8.2e+275) {
		tmp = z * (y / z);
	} else {
		tmp = x * -(y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.7e+26:
		tmp = y + (x / z)
	elif y <= 3.7e+112:
		tmp = y * (-x / z)
	elif y <= 8.2e+275:
		tmp = z * (y / z)
	else:
		tmp = x * -(y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e+26)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 3.7e+112)
		tmp = Float64(y * Float64(Float64(-x) / z));
	elseif (y <= 8.2e+275)
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x * Float64(-Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.7e+26)
		tmp = y + (x / z);
	elseif (y <= 3.7e+112)
		tmp = y * (-x / z);
	elseif (y <= 8.2e+275)
		tmp = z * (y / z);
	else
		tmp = x * -(y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e+26], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+112], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+275], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x * (-N[(y / z), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.7000000000000001e26

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf 77.2%

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

   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%

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

   if 1.7000000000000001e26 < y < 3.70000000000000004e112

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 64.6%

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

      1. associate-/l* 64.3%

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

      2. mul-1-neg 64.3%

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

      3. unsub-neg 64.3%

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

   5. Simplified 64.3%

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

   6. Taylor expanded in y around inf 64.3%

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

      1. neg-mul-1 64.3%

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

   8. Simplified 64.3%

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

      1. distribute-frac-neg 64.3%

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

      2. distribute-frac-neg2 64.3%

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

      3. associate-*r/ 64.6%

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

      4. distribute-frac-neg2 64.6%

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

      5. associate-*l/ 69.0%

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

      6. *-commutative 69.0%

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

   10. Applied egg-rr 69.0%

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

   if 3.70000000000000004e112 < y < 8.1999999999999994e275

   1. Initial program 70.8%

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

   3. Taylor expanded in z around inf 32.7%

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

   4. Taylor expanded in x around 0 33.4%

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

      1. *-commutative 33.4%

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

   6. Simplified 33.4%

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

      1. associate-/l* 64.4%

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

      2. *-commutative 64.4%

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

   8. Applied egg-rr 64.4%

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

   if 8.1999999999999994e275 < y

   1. Initial program 75.9%

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

   3. Taylor expanded in x around inf 65.5%

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

      1. associate-/l* 73.2%

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

      2. mul-1-neg 73.2%

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

      3. unsub-neg 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around inf 73.2%

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

      1. neg-mul-1 73.2%

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

   8. Simplified 73.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+275}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+14)
   (* y (/ (- z x) z))
   (if (<= y 0.00078) (/ (+ x (* y (- z x))) z) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+14) {
		tmp = y * ((z - x) / z);
	} else if (y <= 0.00078) {
		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 <= (-4d+14)) then
        tmp = y * ((z - x) / z)
    else if (y <= 0.00078d0) 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 <= -4e+14) {
		tmp = y * ((z - x) / z);
	} else if (y <= 0.00078) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+14:
		tmp = y * ((z - x) / z)
	elif y <= 0.00078:
		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 <= -4e+14)
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (y <= 0.00078)
		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 <= -4e+14)
		tmp = y * ((z - x) / z);
	elseif (y <= 0.00078)
		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, -4e+14], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00078], 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 < -4e14

   1. Initial program 66.9%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -4e14 < y < 7.79999999999999986e-4

   1. Initial program 99.9%

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

   if 7.79999999999999986e-4 < y

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0 95.7%

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

   4. Taylor expanded in y around inf 91.8%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

      3. distribute-rgt-neg-in 95.7%

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

      4. distribute-frac-neg2 95.7%

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

   6. Simplified 95.7%

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

      1. *-commutative 95.7%

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

      2. add-sqr-sqrt 46.3%

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

      3. sqrt-unprod 65.0%

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

      4. sqr-neg 65.0%

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

      5. sqrt-unprod 25.2%

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

      6. add-sqr-sqrt 45.7%

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

      7. cancel-sign-sub 45.7%

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

      8. distribute-frac-neg2 45.7%

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

      9. associate-*l/ 43.3%

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

      10. *-commutative 43.3%

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

      11. add-sqr-sqrt 20.4%

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

      12. sqrt-unprod 62.8%

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

      13. sqr-neg 62.8%

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

      14. sqrt-unprod 46.7%

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

      15. add-sqr-sqrt 91.8%

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

      16. associate-*l/ 100.0%

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

      17. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -370000000000 \lor \neg \left(y \leq 0.00078\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -370000000000.0) (not (<= y 0.00078)))
   (* y (/ (- z x) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -370000000000.0) || !(y <= 0.00078)) {
		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 <= (-370000000000.0d0)) .or. (.not. (y <= 0.00078d0))) 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 <= -370000000000.0) || !(y <= 0.00078)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -370000000000.0) or not (y <= 0.00078):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -370000000000.0) || !(y <= 0.00078))
		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 <= -370000000000.0) || ~((y <= 0.00078)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -370000000000.0], N[Not[LessEqual[y, 0.00078]], $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 < -3.7e11 or 7.79999999999999986e-4 < y

   1. Initial program 74.2%

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

   3. Taylor expanded in y around inf 74.2%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -3.7e11 < y < 7.79999999999999986e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 99.1%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000000000 \lor \neg \left(y \leq 0.00078\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+101} \lor \neg \left(x \leq 2.05 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.65e+101) (not (<= x 2.05e-19)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.65e+101) || !(x <= 2.05e-19)) {
		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 <= (-2.65d+101)) .or. (.not. (x <= 2.05d-19))) 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 <= -2.65e+101) || !(x <= 2.05e-19)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.65e+101) or not (x <= 2.05e-19):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.65e+101) || !(x <= 2.05e-19))
		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 <= -2.65e+101) || ~((x <= 2.05e-19)))
		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, -2.65e+101], N[Not[LessEqual[x, 2.05e-19]], $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 < -2.65000000000000003e101 or 2.04999999999999993e-19 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 82.6%

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

      1. associate-/l* 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

   5. Simplified 85.7%

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

   if -2.65000000000000003e101 < x < 2.04999999999999993e-19

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf 72.1%

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

   4. Taylor expanded in x around 0 86.7%

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

      1. +-commutative 86.7%

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

   6. Simplified 86.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+101} \lor \neg \left(x \leq 2.05 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -370000000000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -370000000000.0)
   (* y (/ (- z x) z))
   (if (<= y 0.00078) (+ y (/ x z)) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -370000000000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 0.00078) {
		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 <= (-370000000000.0d0)) then
        tmp = y * ((z - x) / z)
    else if (y <= 0.00078d0) 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 <= -370000000000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 0.00078) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -370000000000.0:
		tmp = y * ((z - x) / z)
	elif y <= 0.00078:
		tmp = y + (x / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -370000000000.0)
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (y <= 0.00078)
		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 <= -370000000000.0)
		tmp = y * ((z - x) / z);
	elseif (y <= 0.00078)
		tmp = y + (x / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -370000000000.0], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00078], 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 < -3.7e11

   1. Initial program 66.9%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -3.7e11 < y < 7.79999999999999986e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 99.1%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   if 7.79999999999999986e-4 < y

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0 95.7%

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

   4. Taylor expanded in y around inf 91.8%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

      3. distribute-rgt-neg-in 95.7%

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

      4. distribute-frac-neg2 95.7%

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

   6. Simplified 95.7%

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

      1. *-commutative 95.7%

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

      2. add-sqr-sqrt 46.3%

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

      3. sqrt-unprod 65.0%

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

      4. sqr-neg 65.0%

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

      5. sqrt-unprod 25.2%

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

      6. add-sqr-sqrt 45.7%

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

      7. cancel-sign-sub 45.7%

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

      8. distribute-frac-neg2 45.7%

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

      9. associate-*l/ 43.3%

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

      10. *-commutative 43.3%

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

      11. add-sqr-sqrt 20.4%

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

      12. sqrt-unprod 62.8%

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

      13. sqr-neg 62.8%

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

      14. sqrt-unprod 46.7%

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

      15. add-sqr-sqrt 91.8%

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

      16. associate-*l/ 100.0%

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

      17. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000000000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \left(\frac{1}{z} - \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-286)
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (+ y (* x (- (/ 1.0 z) (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-286) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else {
		tmp = y + (x * ((1.0 / 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 <= 5.2d-286) then
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    else
        tmp = y + (x * ((1.0d0 / z) - (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-286) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else {
		tmp = y + (x * ((1.0 / z) - (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.2e-286:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	else:
		tmp = y + (x * ((1.0 / z) - (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-286)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	else
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 / z) - Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-286)
		tmp = (x / z) + (y * (1.0 - (x / z)));
	else
		tmp = y + (x * ((1.0 / z) - (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e-286], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * N[(N[(1.0 / z), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999999e-286

   1. Initial program 83.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 5.1999999999999999e-286 < y

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 97.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \left(\frac{1}{z} - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-106} \lor \neg \left(y \leq 4.8 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e-106) (not (<= y 4.8e-21))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e-106) || !(y <= 4.8e-21)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3d-106)) .or. (.not. (y <= 4.8d-21))) then
        tmp = z * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e-106) || !(y <= 4.8e-21)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3e-106) or not (y <= 4.8e-21):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e-106) || !(y <= 4.8e-21))
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3e-106) || ~((y <= 4.8e-21)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e-106], N[Not[LessEqual[y, 4.8e-21]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000019e-106 or 4.7999999999999999e-21 < y

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf 46.4%

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

   4. Taylor expanded in x around 0 36.8%

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

      1. *-commutative 36.8%

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

   6. Simplified 36.8%

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

      1. associate-/l* 55.1%

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

      2. *-commutative 55.1%

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

   8. Applied egg-rr 55.1%

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

   if -3.00000000000000019e-106 < y < 4.7999999999999999e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-106} \lor \neg \left(y \leq 4.8 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-106}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e-106) y (if (<= y 1.4e-19) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e-106) {
		tmp = y;
	} else if (y <= 1.4e-19) {
		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 <= (-3d-106)) then
        tmp = y
    else if (y <= 1.4d-19) 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 <= -3e-106) {
		tmp = y;
	} else if (y <= 1.4e-19) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e-106:
		tmp = y
	elif y <= 1.4e-19:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e-106)
		tmp = y;
	elseif (y <= 1.4e-19)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e-106)
		tmp = y;
	elseif (y <= 1.4e-19)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e-106], y, If[LessEqual[y, 1.4e-19], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000019e-106 or 1.40000000000000001e-19 < y

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 54.3%

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

   if -3.00000000000000019e-106 < y < 1.40000000000000001e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.2%

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

Alternative 10: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+31) (+ y (/ x z)) (* y (/ (- x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+31) {
		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 <= 1.25d+31) 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 <= 1.25e+31) {
		tmp = y + (x / z);
	} else {
		tmp = y * (-x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.25e+31:
		tmp = y + (x / z)
	else:
		tmp = y * (-x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e+31)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(Float64(-x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.25e+31)
		tmp = y + (x / z);
	else
		tmp = y * (-x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e+31], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000007e31

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf 77.2%

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

   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%

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

   if 1.25000000000000007e31 < y

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf 50.6%

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

      1. associate-/l* 53.3%

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

      2. mul-1-neg 53.3%

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

      3. unsub-neg 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around inf 53.3%

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

      1. neg-mul-1 53.3%

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

   8. Simplified 53.3%

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

      1. distribute-frac-neg 53.3%

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

      2. distribute-frac-neg2 53.3%

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

      3. associate-*r/ 50.6%

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

      4. distribute-frac-neg2 50.6%

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

      5. associate-*l/ 54.7%

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

      6. *-commutative 54.7%

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

   10. Applied egg-rr 54.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 86.4%

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

3. Taylor expanded in z around inf 66.1%

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

4. Taylor expanded in x around 0 77.2%

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

   1. +-commutative 77.2%

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

6. Simplified 77.2%

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

7. Final simplification 77.2%

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

Alternative 12: 40.1% 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 86.4%

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

3. Taylor expanded in x around 0 44.6%

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

Developer Target 1: 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 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))