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

Percentage Accurate: 88.6% → 98.7%

Time: 7.5s

Alternatives: 10

Speedup: 0.6×

• 20.5% 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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+21} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.02e+21) (not (<= y 1.0)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.02e+21) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.02d+21)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.02e+21) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.02e+21) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.02e+21) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.02e+21) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.02e+21], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e21 or 1 < y

   1. Initial program 75.4%

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

   3. Taylor expanded in y around inf 74.4%

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

      1. associate-/l* 98.9%

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

      2. div-sub 99.0%

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

      3. *-inverses 99.0%

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

   5. Simplified 99.0%

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

   if -1.02e21 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 87.5%

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

   4. Taylor expanded in y around 0 96.9%

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

      1. sub-neg 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-rgt-in 96.9%

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

      4. *-lft-identity 96.9%

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

      5. associate-+l+ 96.9%

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

      6. distribute-frac-neg 96.9%

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

      7. mul-1-neg 96.9%

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

      8. *-commutative 96.9%

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

      9. associate-*l* 96.9%

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

      10. neg-mul-1 96.9%

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

      11. *-rgt-identity 96.9%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 100.0%

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

      16. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 99.0%

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

   8. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+21} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+46} \lor \neg \left(x \leq 7.5 \cdot 10^{+50}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.2e+46) (not (<= x 7.5e+50)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.2e+46) || !(x <= 7.5e+50)) {
		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 <= (-3.2d+46)) .or. (.not. (x <= 7.5d+50))) 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 <= -3.2e+46) || !(x <= 7.5e+50)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.2e+46) or not (x <= 7.5e+50):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.2e+46) || !(x <= 7.5e+50))
		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 <= -3.2e+46) || ~((x <= 7.5e+50)))
		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, -3.2e+46], N[Not[LessEqual[x, 7.5e+50]], $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 < -3.1999999999999998e46 or 7.4999999999999999e50 < x

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 83.1%

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

      1. associate-/l* 90.5%

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

      2. mul-1-neg 90.5%

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

      3. unsub-neg 90.5%

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

   5. Simplified 90.5%

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

   if -3.1999999999999998e46 < x < 7.4999999999999999e50

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf 88.1%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*l* 100.0%

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

      10. neg-mul-1 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 100.0%

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

      16. associate-*r/ 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in y around 0 85.9%

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

   8. Taylor expanded in y around 0 86.0%

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

      1. +-commutative 86.0%

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

   10. Simplified 86.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+46} \lor \neg \left(x \leq 7.5 \cdot 10^{+50}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+171} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.15e+171) (not (<= y 1.0))) (* z (/ y z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.15e+171) || !(y <= 1.0)) {
		tmp = z * (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 ((y <= (-2.15d+171)) .or. (.not. (y <= 1.0d0))) then
        tmp = z * (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 ((y <= -2.15e+171) || !(y <= 1.0)) {
		tmp = z * (y / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.15e+171) or not (y <= 1.0):
		tmp = z * (y / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.15e+171) || !(y <= 1.0))
		tmp = Float64(z * Float64(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 ((y <= -2.15e+171) || ~((y <= 1.0)))
		tmp = z * (y / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.15e+171], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15000000000000004e171 or 1 < y

   1. Initial program 76.5%

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

   3. Taylor expanded in y around inf 76.5%

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

   4. Taylor expanded in z around inf 30.4%

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

      1. *-commutative 30.4%

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

      2. associate-/l* 57.7%

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

   6. Applied egg-rr 57.7%

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

   if -2.15000000000000004e171 < y < 1

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 84.5%

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

   4. Taylor expanded in y around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-rgt-in 97.5%

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

      4. *-lft-identity 97.5%

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

      5. associate-+l+ 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. mul-1-neg 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-*l* 97.5%

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

      10. neg-mul-1 97.5%

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

      11. *-rgt-identity 97.5%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 98.8%

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

      16. associate-*r/ 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in y around 0 91.1%

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

   8. Taylor expanded in y around 0 91.4%

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

      1. +-commutative 91.4%

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

   10. Simplified 91.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+171} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-67} \lor \neg \left(y \leq 2.3 \cdot 10^{-17}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e-67) (not (<= y 2.3e-17))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e-67) || !(y <= 2.3e-17)) {
		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 <= (-1.25d-67)) .or. (.not. (y <= 2.3d-17))) 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 <= -1.25e-67) || !(y <= 2.3e-17)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e-67) or not (y <= 2.3e-17):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e-67) || !(y <= 2.3e-17))
		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 <= -1.25e-67) || ~((y <= 2.3e-17)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e-67], N[Not[LessEqual[y, 2.3e-17]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e-67 or 2.30000000000000009e-17 < y

   1. Initial program 79.4%

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

   3. Taylor expanded in y around inf 79.4%

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

   4. Taylor expanded in z around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/l* 55.8%

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

   6. Applied egg-rr 55.8%

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

   if -1.25e-67 < y < 2.30000000000000009e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 76.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-67} \lor \neg \left(y \leq 2.3 \cdot 10^{-17}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+121}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+121)
   (* y (/ x (- z)))
   (if (<= y 1.0) (+ y (/ x z)) (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+121) {
		tmp = y * (x / -z);
	} else if (y <= 1.0) {
		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 <= (-2.1d+121)) then
        tmp = y * (x / -z)
    else if (y <= 1.0d0) 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 <= -2.1e+121) {
		tmp = y * (x / -z);
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+121:
		tmp = y * (x / -z)
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+121)
		tmp = Float64(y * Float64(x / Float64(-z)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+121)
		tmp = y * (x / -z);
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+121], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1000000000000002e121

   1. Initial program 69.5%

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

   3. Taylor expanded in y around inf 69.5%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 66.3%

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

      1. mul-1-neg 66.3%

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

      2. distribute-frac-neg 66.3%

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

   8. Simplified 66.3%

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

   if -2.1000000000000002e121 < y < 1

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf 85.8%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. *-lft-identity 97.3%

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

      5. associate-+l+ 97.3%

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

      6. distribute-frac-neg 97.3%

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

      7. mul-1-neg 97.3%

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

      8. *-commutative 97.3%

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

      9. associate-*l* 97.3%

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

      10. neg-mul-1 97.3%

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

      11. *-rgt-identity 97.3%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 99.4%

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

      16. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 94.2%

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

   8. Taylor expanded in y around 0 94.4%

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

      1. +-commutative 94.4%

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

   10. Simplified 94.4%

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

   if 1 < y

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf 79.5%

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

   4. Taylor expanded in y around 0 89.0%

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

      1. sub-neg 89.0%

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

      2. distribute-frac-neg 89.0%

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

      3. distribute-rgt-in 89.0%

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

      4. *-lft-identity 89.0%

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

      5. associate-+l+ 89.0%

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

      6. distribute-frac-neg 89.0%

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

      7. mul-1-neg 89.0%

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

      8. *-commutative 89.0%

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

      9. associate-*l* 89.0%

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

      10. neg-mul-1 89.0%

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

      11. *-rgt-identity 89.0%

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

      12. distribute-lft-in 99.9%

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

      13. +-commutative 99.9%

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

      14. sub-neg 99.9%

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

      15. associate-*l/ 94.1%

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

      16. associate-*r/ 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in y around 0 46.5%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 37.2%

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

      5. add-sqr-sqrt 58.9%

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

      6. cancel-sign-sub-inv 58.9%

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

      7. un-div-inv 58.9%

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

   9. Applied egg-rr 58.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+121}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e+147)
   (* x (/ y (- z)))
   (if (<= y 1.0) (+ y (/ x z)) (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+147) {
		tmp = x * (y / -z);
	} else if (y <= 1.0) {
		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 <= (-1.55d+147)) then
        tmp = x * (y / -z)
    else if (y <= 1.0d0) 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 <= -1.55e+147) {
		tmp = x * (y / -z);
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e+147:
		tmp = x * (y / -z)
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e+147)
		tmp = Float64(x * Float64(y / Float64(-z)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e+147)
		tmp = x * (y / -z);
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e+147], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e147

   1. Initial program 70.7%

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

   3. Taylor expanded in x around inf 57.1%

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

      1. associate-/l* 67.4%

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

      2. mul-1-neg 67.4%

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

      3. unsub-neg 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in y around inf 57.1%

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

      1. mul-1-neg 57.1%

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

      2. associate-/l* 67.4%

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

      3. distribute-rgt-neg-in 67.4%

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

      4. distribute-neg-frac 67.4%

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

   8. Simplified 67.4%

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

   if -1.55e147 < y < 1

   1. Initial program 95.1%

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

   3. Taylor expanded in y around inf 84.7%

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

   4. Taylor expanded in y around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-rgt-in 97.5%

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

      4. *-lft-identity 97.5%

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

      5. associate-+l+ 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. mul-1-neg 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-*l* 97.5%

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

      10. neg-mul-1 97.5%

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

      11. *-rgt-identity 97.5%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 98.8%

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

      16. associate-*r/ 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in y around 0 92.1%

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

   8. Taylor expanded in y around 0 92.4%

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

      1. +-commutative 92.4%

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

   10. Simplified 92.4%

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

   if 1 < y

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf 79.5%

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

   4. Taylor expanded in y around 0 89.0%

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

      1. sub-neg 89.0%

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

      2. distribute-frac-neg 89.0%

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

      3. distribute-rgt-in 89.0%

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

      4. *-lft-identity 89.0%

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

      5. associate-+l+ 89.0%

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

      6. distribute-frac-neg 89.0%

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

      7. mul-1-neg 89.0%

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

      8. *-commutative 89.0%

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

      9. associate-*l* 89.0%

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

      10. neg-mul-1 89.0%

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

      11. *-rgt-identity 89.0%

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

      12. distribute-lft-in 99.9%

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

      13. +-commutative 99.9%

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

      14. sub-neg 99.9%

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

      15. associate-*l/ 94.1%

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

      16. associate-*r/ 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in y around 0 46.5%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 37.2%

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

      5. add-sqr-sqrt 58.9%

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

      6. cancel-sign-sub-inv 58.9%

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

      7. un-div-inv 58.9%

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

   9. Applied egg-rr 58.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+170}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+170) (* z (/ y z)) (if (<= y 1.0) (+ y (/ x z)) (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+170) {
		tmp = z * (y / z);
	} else if (y <= 1.0) {
		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 <= (-4d+170)) then
        tmp = z * (y / z)
    else if (y <= 1.0d0) 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 <= -4e+170) {
		tmp = z * (y / z);
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+170:
		tmp = z * (y / z)
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+170)
		tmp = Float64(z * Float64(y / z));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e+170)
		tmp = z * (y / z);
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+170], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000014e170

   1. Initial program 70.1%

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

   3. Taylor expanded in y around inf 70.1%

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

   4. Taylor expanded in z around inf 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/l* 65.8%

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

   6. Applied egg-rr 65.8%

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

   if -4.00000000000000014e170 < y < 1

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 84.5%

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

   4. Taylor expanded in y around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-rgt-in 97.5%

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

      4. *-lft-identity 97.5%

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

      5. associate-+l+ 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. mul-1-neg 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-*l* 97.5%

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

      10. neg-mul-1 97.5%

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

      11. *-rgt-identity 97.5%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 98.8%

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

      16. associate-*r/ 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in y around 0 91.1%

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

   8. Taylor expanded in y around 0 91.4%

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

      1. +-commutative 91.4%

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

   10. Simplified 91.4%

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

   if 1 < y

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf 79.5%

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

   4. Taylor expanded in y around 0 89.0%

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

      1. sub-neg 89.0%

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

      2. distribute-frac-neg 89.0%

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

      3. distribute-rgt-in 89.0%

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

      4. *-lft-identity 89.0%

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

      5. associate-+l+ 89.0%

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

      6. distribute-frac-neg 89.0%

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

      7. mul-1-neg 89.0%

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

      8. *-commutative 89.0%

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

      9. associate-*l* 89.0%

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

      10. neg-mul-1 89.0%

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

      11. *-rgt-identity 89.0%

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

      12. distribute-lft-in 99.9%

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

      13. +-commutative 99.9%

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

      14. sub-neg 99.9%

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

      15. associate-*l/ 94.1%

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

      16. associate-*r/ 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in y around 0 46.5%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 37.2%

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

      5. add-sqr-sqrt 58.9%

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

      6. cancel-sign-sub-inv 58.9%

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

      7. un-div-inv 58.9%

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

   9. Applied egg-rr 58.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+170}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 430000:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 430000.0) (+ y (* x (/ (- 1.0 y) z))) (* y (- 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 430000.0) {
		tmp = y + (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 <= 430000.0d0) then
        tmp = y + (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 <= 430000.0) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 430000.0:
		tmp = y + (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 <= 430000.0)
		tmp = Float64(y + Float64(x * Float64(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 <= 430000.0)
		tmp = y + (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, 430000.0], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e5

   1. Initial program 90.9%

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

   3. Taylor expanded in y around inf 82.4%

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

   4. Taylor expanded in y around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-rgt-in 97.9%

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

      4. *-lft-identity 97.9%

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

      5. associate-+l+ 97.9%

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

      6. distribute-frac-neg 97.9%

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

      7. mul-1-neg 97.9%

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

      8. *-commutative 97.9%

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

      9. associate-*l* 97.9%

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

      10. neg-mul-1 97.9%

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

      11. *-rgt-identity 97.9%

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

      12. distribute-lft-in 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*l/ 96.7%

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

      16. associate-*r/ 98.8%

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

   6. Simplified 98.8%

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

   if 4.3e5 < y

   1. Initial program 78.8%

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

   3. Taylor expanded in y around inf 78.8%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

Alternative 9: 59.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-68}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-68) y (if (<= y 1.05e-13) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e-68) {
		tmp = y;
	} else if (y <= 1.05e-13) {
		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 <= (-6d-68)) then
        tmp = y
    else if (y <= 1.05d-13) 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 <= -6e-68) {
		tmp = y;
	} else if (y <= 1.05e-13) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e-68:
		tmp = y
	elif y <= 1.05e-13:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e-68)
		tmp = y;
	elseif (y <= 1.05e-13)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6e-68)
		tmp = y;
	elseif (y <= 1.05e-13)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e-68], y, If[LessEqual[y, 1.05e-13], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -6e-68 or 1.04999999999999994e-13 < y

   1. Initial program 79.4%

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

   3. Taylor expanded in x around 0 47.6%

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

   if -6e-68 < y < 1.04999999999999994e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 76.4%

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

Alternative 10: 41.4% 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 88.0%

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

3. Taylor expanded in x around 0 38.4%

   \[\leadsto \color{blue}{y} \]
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 2024185 
(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))