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

Percentage Accurate: 88.0% → 97.8%

Time: 6.8s

Alternatives: 11

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+27}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+27) (- y (/ y (/ z x))) (+ y (* x (/ (- 1.0 y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+27) {
		tmp = y - (y / (z / x));
	} 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 (y <= (-4d+27)) then
        tmp = y - (y / (z / x))
    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 (y <= -4e+27) {
		tmp = y - (y / (z / x));
	} else {
		tmp = y + (x * ((1.0 - y) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+27:
		tmp = y - (y / (z / x))
	else:
		tmp = y + (x * ((1.0 - y) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+27)
		tmp = Float64(y - Float64(y / Float64(z / x)));
	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 (y <= -4e+27)
		tmp = y - (y / (z / x));
	else
		tmp = y + (x * ((1.0 - y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+27], N[(y - N[(y / N[(z / x), $MachinePrecision]), $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 y < -4.0000000000000001e27

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf 77.0%

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

      1. associate-+r+ 77.0%

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

      2. +-commutative 77.0%

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

      3. mul-1-neg 77.0%

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

      4. unsub-neg 77.0%

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

      5. div-sub 77.0%

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

   5. Simplified 77.0%

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

   6. Taylor expanded in x around 0 88.0%

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

      1. div-sub 88.0%

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

   8. Simplified 88.0%

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

   9. Taylor expanded in y around inf 87.8%

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

       1. associate-*l/ 99.9%

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

       2. associate-*l* 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-*l* 99.9%

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

       5. neg-mul-1 99.9%

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

   11. Simplified 99.9%

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

       1. distribute-rgt-neg-out 99.9%

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

       2. distribute-lft-neg-in 99.9%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 22.4%

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

       5. sqr-neg 22.4%

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

       6. sqrt-unprod 52.7%

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

       7. add-sqr-sqrt 52.7%

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

       8. cancel-sign-sub-inv 52.7%

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

       9. *-commutative 52.7%

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

       10. clear-num 52.7%

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

       11. un-div-inv 52.7%

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

       12. add-sqr-sqrt 52.7%

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

       13. sqrt-unprod 22.4%

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

       14. sqr-neg 22.4%

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

       15. sqrt-unprod 0.0%

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

       16. add-sqr-sqrt 99.9%

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

   13. Applied egg-rr 99.9%

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

   if -4.0000000000000001e27 < y

   1. Initial program 91.8%

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

   3. Taylor expanded in x around inf 90.6%

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

      1. associate-+r+ 90.6%

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

      2. +-commutative 90.6%

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

      3. mul-1-neg 90.6%

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

      4. unsub-neg 90.6%

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

      5. div-sub 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in x around 0 97.8%

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

      1. div-sub 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+27}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{-z}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+229}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -2600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-14}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+228}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x (- z)))))
   (if (<= y -5.2e+229)
     y
     (if (<= y -7.2e+199)
       t_0
       (if (<= y -5e+93)
         (/ z (/ z y))
         (if (<= y -2600000000.0)
           t_0
           (if (<= y -1.45e-14)
             y
             (if (<= y 9.5e-15)
               (/ x z)
               (if (<= y 3.6e+228) y (* x (/ y (- z))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / -z);
	double tmp;
	if (y <= -5.2e+229) {
		tmp = y;
	} else if (y <= -7.2e+199) {
		tmp = t_0;
	} else if (y <= -5e+93) {
		tmp = z / (z / y);
	} else if (y <= -2600000000.0) {
		tmp = t_0;
	} else if (y <= -1.45e-14) {
		tmp = y;
	} else if (y <= 9.5e-15) {
		tmp = x / z;
	} else if (y <= 3.6e+228) {
		tmp = y;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (x / -z)
    if (y <= (-5.2d+229)) then
        tmp = y
    else if (y <= (-7.2d+199)) then
        tmp = t_0
    else if (y <= (-5d+93)) then
        tmp = z / (z / y)
    else if (y <= (-2600000000.0d0)) then
        tmp = t_0
    else if (y <= (-1.45d-14)) then
        tmp = y
    else if (y <= 9.5d-15) then
        tmp = x / z
    else if (y <= 3.6d+228) then
        tmp = y
    else
        tmp = x * (y / -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / -z);
	double tmp;
	if (y <= -5.2e+229) {
		tmp = y;
	} else if (y <= -7.2e+199) {
		tmp = t_0;
	} else if (y <= -5e+93) {
		tmp = z / (z / y);
	} else if (y <= -2600000000.0) {
		tmp = t_0;
	} else if (y <= -1.45e-14) {
		tmp = y;
	} else if (y <= 9.5e-15) {
		tmp = x / z;
	} else if (y <= 3.6e+228) {
		tmp = y;
	} else {
		tmp = x * (y / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / -z)
	tmp = 0
	if y <= -5.2e+229:
		tmp = y
	elif y <= -7.2e+199:
		tmp = t_0
	elif y <= -5e+93:
		tmp = z / (z / y)
	elif y <= -2600000000.0:
		tmp = t_0
	elif y <= -1.45e-14:
		tmp = y
	elif y <= 9.5e-15:
		tmp = x / z
	elif y <= 3.6e+228:
		tmp = y
	else:
		tmp = x * (y / -z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / Float64(-z)))
	tmp = 0.0
	if (y <= -5.2e+229)
		tmp = y;
	elseif (y <= -7.2e+199)
		tmp = t_0;
	elseif (y <= -5e+93)
		tmp = Float64(z / Float64(z / y));
	elseif (y <= -2600000000.0)
		tmp = t_0;
	elseif (y <= -1.45e-14)
		tmp = y;
	elseif (y <= 9.5e-15)
		tmp = Float64(x / z);
	elseif (y <= 3.6e+228)
		tmp = y;
	else
		tmp = Float64(x * Float64(y / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / -z);
	tmp = 0.0;
	if (y <= -5.2e+229)
		tmp = y;
	elseif (y <= -7.2e+199)
		tmp = t_0;
	elseif (y <= -5e+93)
		tmp = z / (z / y);
	elseif (y <= -2600000000.0)
		tmp = t_0;
	elseif (y <= -1.45e-14)
		tmp = y;
	elseif (y <= 9.5e-15)
		tmp = x / z;
	elseif (y <= 3.6e+228)
		tmp = y;
	else
		tmp = x * (y / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+229], y, If[LessEqual[y, -7.2e+199], t$95$0, If[LessEqual[y, -5e+93], N[(z / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2600000000.0], t$95$0, If[LessEqual[y, -1.45e-14], y, If[LessEqual[y, 9.5e-15], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.6e+228], y, N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.2e229 or -2.6e9 < y < -1.4500000000000001e-14 or 9.5000000000000005e-15 < y < 3.6e228

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0 62.6%

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

   if -5.2e229 < y < -7.20000000000000002e199 or -5.0000000000000001e93 < y < -2.6e9

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf 62.5%

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

      1. associate-/l* 59.5%

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

      2. mul-1-neg 59.5%

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

      3. unsub-neg 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in y around inf 61.0%

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

      1. associate-*l/ 98.4%

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

      2. associate-*l* 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-*l* 98.4%

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

      5. neg-mul-1 98.4%

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

   8. Simplified 66.2%

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

   if -7.20000000000000002e199 < y < -5.0000000000000001e93

   1. Initial program 61.8%

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

   3. Taylor expanded in x around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-/l* 71.6%

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

   5. Applied egg-rr 71.6%

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

      1. clear-num 71.5%

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

      2. un-div-inv 71.7%

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

   7. Applied egg-rr 71.7%

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

   if -1.4500000000000001e-14 < y < 9.5000000000000005e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.6%

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

   if 3.6e228 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in x around inf 62.7%

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

      1. associate-/l* 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in y around inf 68.2%

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

      1. neg-mul-1 68.2%

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

      2. distribute-neg-frac2 68.2%

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

   8. Simplified 68.2%

      \[\leadsto x \cdot \color{blue}{\frac{y}{-z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+229}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+199}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -2600000000:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-14}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+228}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{-z}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -30000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+226}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y (- z)))))
   (if (<= y -4.8e+93)
     y
     (if (<= y -30000000000.0)
       t_0
       (if (<= y -5.6e-16)
         y
         (if (<= y 1.3e-25) (/ x z) (if (<= y 1.75e+226) y t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y / -z);
	double tmp;
	if (y <= -4.8e+93) {
		tmp = y;
	} else if (y <= -30000000000.0) {
		tmp = t_0;
	} else if (y <= -5.6e-16) {
		tmp = y;
	} else if (y <= 1.3e-25) {
		tmp = x / z;
	} else if (y <= 1.75e+226) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x * (y / -z)
    if (y <= (-4.8d+93)) then
        tmp = y
    else if (y <= (-30000000000.0d0)) then
        tmp = t_0
    else if (y <= (-5.6d-16)) then
        tmp = y
    else if (y <= 1.3d-25) then
        tmp = x / z
    else if (y <= 1.75d+226) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y / -z);
	double tmp;
	if (y <= -4.8e+93) {
		tmp = y;
	} else if (y <= -30000000000.0) {
		tmp = t_0;
	} else if (y <= -5.6e-16) {
		tmp = y;
	} else if (y <= 1.3e-25) {
		tmp = x / z;
	} else if (y <= 1.75e+226) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y / -z)
	tmp = 0
	if y <= -4.8e+93:
		tmp = y
	elif y <= -30000000000.0:
		tmp = t_0
	elif y <= -5.6e-16:
		tmp = y
	elif y <= 1.3e-25:
		tmp = x / z
	elif y <= 1.75e+226:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y / Float64(-z)))
	tmp = 0.0
	if (y <= -4.8e+93)
		tmp = y;
	elseif (y <= -30000000000.0)
		tmp = t_0;
	elseif (y <= -5.6e-16)
		tmp = y;
	elseif (y <= 1.3e-25)
		tmp = Float64(x / z);
	elseif (y <= 1.75e+226)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y / -z);
	tmp = 0.0;
	if (y <= -4.8e+93)
		tmp = y;
	elseif (y <= -30000000000.0)
		tmp = t_0;
	elseif (y <= -5.6e-16)
		tmp = y;
	elseif (y <= 1.3e-25)
		tmp = x / z;
	elseif (y <= 1.75e+226)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+93], y, If[LessEqual[y, -30000000000.0], t$95$0, If[LessEqual[y, -5.6e-16], y, If[LessEqual[y, 1.3e-25], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.75e+226], y, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000021e93 or -3e10 < y < -5.6000000000000003e-16 or 1.3e-25 < y < 1.7499999999999999e226

   1. Initial program 77.1%

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

   3. Taylor expanded in x around 0 60.6%

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

   if -4.80000000000000021e93 < y < -3e10 or 1.7499999999999999e226 < y

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 60.9%

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

      1. associate-/l* 65.2%

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

      2. mul-1-neg 65.2%

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

      3. unsub-neg 65.2%

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

   5. Simplified 65.2%

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

   6. Taylor expanded in y around inf 64.0%

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

      1. neg-mul-1 64.0%

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

      2. distribute-neg-frac2 64.0%

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

   8. Simplified 64.0%

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

   if -5.6000000000000003e-16 < y < 1.3e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -30000000000:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+226}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1 - y}{z}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+109}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot z}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (- 1.0 y) z))))
   (if (<= z -4.5e+109)
     y
     (if (<= z -3.6e+31)
       t_0
       (if (<= z -2.75e-14) (/ (* y z) z) (if (<= z 3.4e+120) t_0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((1.0 - y) / z);
	double tmp;
	if (z <= -4.5e+109) {
		tmp = y;
	} else if (z <= -3.6e+31) {
		tmp = t_0;
	} else if (z <= -2.75e-14) {
		tmp = (y * z) / z;
	} else if (z <= 3.4e+120) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * ((1.0d0 - y) / z)
    if (z <= (-4.5d+109)) then
        tmp = y
    else if (z <= (-3.6d+31)) then
        tmp = t_0
    else if (z <= (-2.75d-14)) then
        tmp = (y * z) / z
    else if (z <= 3.4d+120) then
        tmp = t_0
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * ((1.0 - y) / z);
	double tmp;
	if (z <= -4.5e+109) {
		tmp = y;
	} else if (z <= -3.6e+31) {
		tmp = t_0;
	} else if (z <= -2.75e-14) {
		tmp = (y * z) / z;
	} else if (z <= 3.4e+120) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * ((1.0 - y) / z)
	tmp = 0
	if z <= -4.5e+109:
		tmp = y
	elif z <= -3.6e+31:
		tmp = t_0
	elif z <= -2.75e-14:
		tmp = (y * z) / z
	elif z <= 3.4e+120:
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(1.0 - y) / z))
	tmp = 0.0
	if (z <= -4.5e+109)
		tmp = y;
	elseif (z <= -3.6e+31)
		tmp = t_0;
	elseif (z <= -2.75e-14)
		tmp = Float64(Float64(y * z) / z);
	elseif (z <= 3.4e+120)
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * ((1.0 - y) / z);
	tmp = 0.0;
	if (z <= -4.5e+109)
		tmp = y;
	elseif (z <= -3.6e+31)
		tmp = t_0;
	elseif (z <= -2.75e-14)
		tmp = (y * z) / z;
	elseif (z <= 3.4e+120)
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+109], y, If[LessEqual[z, -3.6e+31], t$95$0, If[LessEqual[z, -2.75e-14], N[(N[(y * z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 3.4e+120], t$95$0, y]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4999999999999996e109 or 3.39999999999999999e120 < z

   1. Initial program 63.6%

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

   3. Taylor expanded in x around 0 74.5%

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

   if -4.4999999999999996e109 < z < -3.59999999999999996e31 or -2.74999999999999996e-14 < z < 3.39999999999999999e120

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf 77.2%

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

      1. associate-/l* 77.1%

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

      2. mul-1-neg 77.1%

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

      3. unsub-neg 77.1%

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

   5. Simplified 77.1%

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

   if -3.59999999999999996e31 < z < -2.74999999999999996e-14

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 72.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+109}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot z}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+120}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-36} \lor \neg \left(y \leq 9.8 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.5e-36) (not (<= y 9.8e-17))) (* y (- 1.0 (/ x z))) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e-36) || !(y <= 9.8e-17)) {
		tmp = y * (1.0 - (x / 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 <= (-6.5d-36)) .or. (.not. (y <= 9.8d-17))) then
        tmp = y * (1.0d0 - (x / 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 <= -6.5e-36) || !(y <= 9.8e-17)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.5e-36) or not (y <= 9.8e-17):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.5e-36) || !(y <= 9.8e-17))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.5e-36) || ~((y <= 9.8e-17)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.5e-36], N[Not[LessEqual[y, 9.8e-17]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000012e-36 or 9.80000000000000024e-17 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf 77.1%

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

      1. associate-/l* 98.8%

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

      2. div-sub 98.8%

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

      3. *-inverses 98.8%

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

   5. Simplified 98.8%

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

   if -6.50000000000000012e-36 < y < 9.80000000000000024e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-36} \lor \neg \left(y \leq 9.8 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-35)
   (- y (/ y (/ z x)))
   (if (<= y 2.55e-17) (/ x z) (* y (- 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-35) {
		tmp = y - (y / (z / x));
	} else if (y <= 2.55e-17) {
		tmp = x / z;
	} else {
		tmp = y * (1.0 - (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 <= (-1.5d-35)) then
        tmp = y - (y / (z / x))
    else if (y <= 2.55d-17) then
        tmp = x / z
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-35) {
		tmp = y - (y / (z / x));
	} else if (y <= 2.55e-17) {
		tmp = x / z;
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e-35:
		tmp = y - (y / (z / x))
	elif y <= 2.55e-17:
		tmp = x / z
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-35)
		tmp = Float64(y - Float64(y / Float64(z / x)));
	elseif (y <= 2.55e-17)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.5e-35)
		tmp = y - (y / (z / x));
	elseif (y <= 2.55e-17)
		tmp = x / z;
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e-35], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e-17], N[(x / z), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.49999999999999994e-35

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf 80.3%

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

      1. associate-+r+ 80.3%

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

      2. +-commutative 80.3%

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

      3. mul-1-neg 80.3%

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

      4. unsub-neg 80.3%

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

      5. div-sub 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in x around 0 89.8%

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

      1. div-sub 89.8%

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

   8. Simplified 89.8%

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

   9. Taylor expanded in y around inf 88.0%

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

       1. associate-*l/ 98.4%

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

       2. associate-*l* 98.4%

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

       3. *-commutative 98.4%

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

       4. associate-*l* 98.4%

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

       5. neg-mul-1 98.4%

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

   11. Simplified 98.4%

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

       1. distribute-rgt-neg-out 98.4%

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

       2. distribute-lft-neg-in 98.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 27.6%

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

       5. sqr-neg 27.6%

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

       6. sqrt-unprod 53.5%

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

       7. add-sqr-sqrt 53.5%

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

       8. cancel-sign-sub-inv 53.5%

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

       9. *-commutative 53.5%

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

       10. clear-num 53.5%

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

       11. un-div-inv 53.5%

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

       12. add-sqr-sqrt 53.5%

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

       13. sqrt-unprod 27.6%

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

       14. sqr-neg 27.6%

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

       15. sqrt-unprod 0.0%

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

       16. add-sqr-sqrt 98.4%

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

   13. Applied egg-rr 98.4%

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

   if -1.49999999999999994e-35 < y < 2.5500000000000001e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.2%

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

   if 2.5500000000000001e-17 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf 77.7%

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

      1. associate-/l* 99.4%

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

      2. div-sub 99.4%

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

      3. *-inverses 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-32}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{x \cdot \left(1 - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e-32)
   (- y (/ y (/ z x)))
   (if (<= y 2.4e-18) (/ (* x (- 1.0 y)) z) (* y (- 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-32) {
		tmp = y - (y / (z / x));
	} else if (y <= 2.4e-18) {
		tmp = (x * (1.0 - y)) / z;
	} else {
		tmp = y * (1.0 - (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 <= (-8d-32)) then
        tmp = y - (y / (z / x))
    else if (y <= 2.4d-18) then
        tmp = (x * (1.0d0 - y)) / z
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e-32) {
		tmp = y - (y / (z / x));
	} else if (y <= 2.4e-18) {
		tmp = (x * (1.0 - y)) / z;
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8e-32:
		tmp = y - (y / (z / x))
	elif y <= 2.4e-18:
		tmp = (x * (1.0 - y)) / z
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e-32)
		tmp = Float64(y - Float64(y / Float64(z / x)));
	elseif (y <= 2.4e-18)
		tmp = Float64(Float64(x * Float64(1.0 - y)) / z);
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8e-32)
		tmp = y - (y / (z / x));
	elseif (y <= 2.4e-18)
		tmp = (x * (1.0 - y)) / z;
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e-32], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-18], N[(N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000045e-32

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf 80.3%

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

      1. associate-+r+ 80.3%

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

      2. +-commutative 80.3%

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

      3. mul-1-neg 80.3%

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

      4. unsub-neg 80.3%

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

      5. div-sub 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in x around 0 89.8%

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

      1. div-sub 89.8%

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

   8. Simplified 89.8%

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

   9. Taylor expanded in y around inf 88.0%

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

       1. associate-*l/ 98.4%

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

       2. associate-*l* 98.4%

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

       3. *-commutative 98.4%

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

       4. associate-*l* 98.4%

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

       5. neg-mul-1 98.4%

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

   11. Simplified 98.4%

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

       1. distribute-rgt-neg-out 98.4%

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

       2. distribute-lft-neg-in 98.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 27.6%

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

       5. sqr-neg 27.6%

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

       6. sqrt-unprod 53.5%

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

       7. add-sqr-sqrt 53.5%

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

       8. cancel-sign-sub-inv 53.5%

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

       9. *-commutative 53.5%

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

       10. clear-num 53.5%

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

       11. un-div-inv 53.5%

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

       12. add-sqr-sqrt 53.5%

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

       13. sqrt-unprod 27.6%

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

       14. sqr-neg 27.6%

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

       15. sqrt-unprod 0.0%

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

       16. add-sqr-sqrt 98.4%

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

   13. Applied egg-rr 98.4%

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

   if -8.00000000000000045e-32 < y < 2.39999999999999994e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

   5. Simplified 80.2%

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

   if 2.39999999999999994e-18 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf 77.7%

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

      1. associate-/l* 99.4%

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

      2. div-sub 99.4%

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

      3. *-inverses 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-32}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{x \cdot \left(1 - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-16} \lor \neg \left(y \leq 1.6 \cdot 10^{-23}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e-16) (not (<= y 1.6e-23))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e-16) || !(y <= 1.6e-23)) {
		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 <= (-4.1d-16)) .or. (.not. (y <= 1.6d-23))) 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 <= -4.1e-16) || !(y <= 1.6e-23)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e-16) or not (y <= 1.6e-23):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e-16) || !(y <= 1.6e-23))
		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 <= -4.1e-16) || ~((y <= 1.6e-23)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e-16], N[Not[LessEqual[y, 1.6e-23]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.10000000000000006e-16 or 1.59999999999999988e-23 < y

   1. Initial program 77.9%

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

   3. Taylor expanded in x around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-/l* 53.5%

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

   5. Applied egg-rr 53.5%

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

   if -4.10000000000000006e-16 < y < 1.59999999999999988e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-16} \lor \neg \left(y \leq 1.6 \cdot 10^{-23}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-13} \lor \neg \left(y \leq 1.08 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7e-13) (not (<= y 1.08e-14))) (/ z (/ z y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e-13) || !(y <= 1.08e-14)) {
		tmp = z / (z / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.7d-13)) .or. (.not. (y <= 1.08d-14))) then
        tmp = z / (z / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e-13) || !(y <= 1.08e-14)) {
		tmp = z / (z / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.7e-13) or not (y <= 1.08e-14):
		tmp = z / (z / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.7e-13) || !(y <= 1.08e-14))
		tmp = Float64(z / Float64(z / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.7e-13) || ~((y <= 1.08e-14)))
		tmp = z / (z / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.7e-13], N[Not[LessEqual[y, 1.08e-14]], $MachinePrecision]], N[(z / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000008e-13 or 1.08000000000000004e-14 < y

   1. Initial program 77.9%

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

   3. Taylor expanded in x around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-/l* 53.5%

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

   5. Applied egg-rr 53.5%

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

      1. clear-num 53.4%

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

      2. un-div-inv 53.6%

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

   7. Applied egg-rr 53.6%

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

   if -1.70000000000000008e-13 < y < 1.08000000000000004e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-13} \lor \neg \left(y \leq 1.08 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e-16) y (if (<= y 5.8e-18) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-16) {
		tmp = y;
	} else if (y <= 5.8e-18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.7d-16)) then
        tmp = y
    else if (y <= 5.8d-18) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-16) {
		tmp = y;
	} else if (y <= 5.8e-18) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.7e-16:
		tmp = y
	elif y <= 5.8e-18:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e-16)
		tmp = y;
	elseif (y <= 5.8e-18)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.7e-16)
		tmp = y;
	elseif (y <= 5.8e-18)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e-16], y, If[LessEqual[y, 5.8e-18], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.69999999999999999e-16 or 5.8e-18 < y

   1. Initial program 77.9%

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

   3. Taylor expanded in x around 0 53.4%

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

   if -2.69999999999999999e-16 < y < 5.8e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.1% 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 87.1%

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

3. Taylor expanded in x around 0 41.0%

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

4. Final simplification 41.0%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 94.3% 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 2024073 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (- (+ y (/ x z)) (/ y (/ z x)))

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