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

Percentage Accurate: 88.4% → 99.9%

Time: 7.8s

Alternatives: 11

Speedup: 1.0×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y + \frac{x}{z} \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ y (* (/ x z) (- 1.0 y))))

C

double code(double x, double y, double z) {
	return y + ((x / z) * (1.0 - 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 + ((x / z) * (1.0d0 - y))
end function

Java

public static double code(double x, double y, double z) {
	return y + ((x / z) * (1.0 - y));
}

Python

def code(x, y, z):
	return y + ((x / z) * (1.0 - y))

Julia

function code(x, y, z)
	return Float64(y + Float64(Float64(x / z) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y + ((x / z) * (1.0 - y));
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in y around 0 96.0%

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

4. Taylor expanded in x around 0 95.4%

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

   1. neg-mul-1 95.4%

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

   2. +-commutative 95.4%

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

   3. sub-neg 95.4%

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

   4. div-sub 95.4%

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

   5. associate-/l* 97.0%

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

   6. associate-*l/ 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

   \[\leadsto y + \frac{x}{z} \cdot \left(1 - y\right) \]
8. Add Preprocessing

Alternative 2: 56.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.65 \cdot 10^{+56} \lor \neg \left(x \leq -1.5 \cdot 10^{-108} \lor \neg \left(x \leq -6.5 \cdot 10^{-124}\right) \land x \leq 1.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.65e+56)
         (not
          (or (<= x -1.5e-108) (and (not (<= x -6.5e-124)) (<= x 1.8e-11)))))
   (/ x z)
   y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.65e+56) || !((x <= -1.5e-108) || (!(x <= -6.5e-124) && (x <= 1.8e-11)))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.65d+56)) .or. (.not. (x <= (-1.5d-108)) .or. (.not. (x <= (-6.5d-124))) .and. (x <= 1.8d-11))) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.65e+56) || !((x <= -1.5e-108) || (!(x <= -6.5e-124) && (x <= 1.8e-11)))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.65e+56) or not ((x <= -1.5e-108) or (not (x <= -6.5e-124) and (x <= 1.8e-11))):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.65e+56) || !((x <= -1.5e-108) || (!(x <= -6.5e-124) && (x <= 1.8e-11))))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.65e+56) || ~(((x <= -1.5e-108) || (~((x <= -6.5e-124)) && (x <= 1.8e-11)))))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.65e+56], N[Not[Or[LessEqual[x, -1.5e-108], And[N[Not[LessEqual[x, -6.5e-124]], $MachinePrecision], LessEqual[x, 1.8e-11]]]], $MachinePrecision]], N[(x / z), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -3.65e56 or -1.49999999999999996e-108 < x < -6.49999999999999988e-124 or 1.79999999999999992e-11 < x

   1. Initial program 91.8%

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

   3. Taylor expanded in y around 0 55.6%

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

   if -3.65e56 < x < -1.49999999999999996e-108 or -6.49999999999999988e-124 < x < 1.79999999999999992e-11

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0 75.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.65 \cdot 10^{+56} \lor \neg \left(x \leq -1.5 \cdot 10^{-108} \lor \neg \left(x \leq -6.5 \cdot 10^{-124}\right) \land x \leq 1.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+103} \lor \neg \left(y \leq 8.5 \cdot 10^{+256}\right):\\ \;\;\;\;-x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+42)
   (+ y (/ x z))
   (if (or (<= y 3.55e+103) (not (<= y 8.5e+256)))
     (- (* x (/ y z)))
     (/ (* y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+42) {
		tmp = y + (x / z);
	} else if ((y <= 3.55e+103) || !(y <= 8.5e+256)) {
		tmp = -(x * (y / z));
	} else {
		tmp = (y * x) / x;
	}
	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 <= 1.3d+42) then
        tmp = y + (x / z)
    else if ((y <= 3.55d+103) .or. (.not. (y <= 8.5d+256))) then
        tmp = -(x * (y / z))
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+42) {
		tmp = y + (x / z);
	} else if ((y <= 3.55e+103) || !(y <= 8.5e+256)) {
		tmp = -(x * (y / z));
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.3e+42:
		tmp = y + (x / z)
	elif (y <= 3.55e+103) or not (y <= 8.5e+256):
		tmp = -(x * (y / z))
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+42)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 3.55e+103) || !(y <= 8.5e+256))
		tmp = Float64(-Float64(x * Float64(y / z)));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.3e+42)
		tmp = y + (x / z);
	elseif ((y <= 3.55e+103) || ~((y <= 8.5e+256)))
		tmp = -(x * (y / z));
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e+42], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.55e+103], N[Not[LessEqual[y, 8.5e+256]], $MachinePrecision]], (-N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.29999999999999995e42

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 98.0%

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

   4. Taylor expanded in x around 0 89.0%

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

   if 1.29999999999999995e42 < y < 3.5500000000000001e103 or 8.5000000000000006e256 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf 74.4%

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

      1. associate-/l* 74.4%

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

      2. mul-1-neg 74.4%

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

      3. unsub-neg 74.4%

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

   5. Simplified 74.4%

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

   6. Taylor expanded in y around inf 74.4%

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

      1. neg-mul-1 74.4%

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

      2. distribute-neg-frac 74.4%

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

   8. Simplified 74.4%

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

   if 3.5500000000000001e103 < y < 8.5000000000000006e256

   1. Initial program 77.0%

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

   3. Taylor expanded in x around inf 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. mul-1-neg 64.0%

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

      4. unsub-neg 64.0%

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

      5. div-sub 64.0%

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

   5. Simplified 64.0%

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

   6. Taylor expanded in z around inf 22.8%

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

      1. associate-*r/ 63.7%

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

   8. Applied egg-rr 63.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+103} \lor \neg \left(y \leq 8.5 \cdot 10^{+256}\right):\\ \;\;\;\;-x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+44}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+102} \lor \neg \left(y \leq 9.4 \cdot 10^{+256}\right):\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+44)
   (+ y (/ x z))
   (if (or (<= y 4.2e+102) (not (<= y 9.4e+256)))
     (* y (/ x (- z)))
     (/ (* y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+44) {
		tmp = y + (x / z);
	} else if ((y <= 4.2e+102) || !(y <= 9.4e+256)) {
		tmp = y * (x / -z);
	} else {
		tmp = (y * x) / x;
	}
	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 <= 2d+44) then
        tmp = y + (x / z)
    else if ((y <= 4.2d+102) .or. (.not. (y <= 9.4d+256))) then
        tmp = y * (x / -z)
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+44) {
		tmp = y + (x / z);
	} else if ((y <= 4.2e+102) || !(y <= 9.4e+256)) {
		tmp = y * (x / -z);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+44:
		tmp = y + (x / z)
	elif (y <= 4.2e+102) or not (y <= 9.4e+256):
		tmp = y * (x / -z)
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+44)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 4.2e+102) || !(y <= 9.4e+256))
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+44)
		tmp = y + (x / z);
	elseif ((y <= 4.2e+102) || ~((y <= 9.4e+256)))
		tmp = y * (x / -z);
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+44], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 4.2e+102], N[Not[LessEqual[y, 9.4e+256]], $MachinePrecision]], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.0000000000000002e44

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 98.0%

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

   4. Taylor expanded in x around 0 89.0%

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

   if 2.0000000000000002e44 < y < 4.20000000000000003e102 or 9.39999999999999935e256 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf 85.5%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. distribute-frac-neg2 79.3%

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

   8. Simplified 79.3%

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

   if 4.20000000000000003e102 < y < 9.39999999999999935e256

   1. Initial program 77.0%

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

   3. Taylor expanded in x around inf 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. mul-1-neg 64.0%

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

      4. unsub-neg 64.0%

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

      5. div-sub 64.0%

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

   5. Simplified 64.0%

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

   6. Taylor expanded in z around inf 22.8%

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

      1. associate-*r/ 63.7%

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

   8. Applied egg-rr 63.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+44}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+102} \lor \neg \left(y \leq 9.4 \cdot 10^{+256}\right):\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+43}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+104} \lor \neg \left(y \leq 8.8 \cdot 10^{+256}\right):\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e+43)
   (+ y (/ x z))
   (if (or (<= y 1.55e+104) (not (<= y 8.8e+256)))
     (/ y (/ (- z) x))
     (/ (* y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e+43) {
		tmp = y + (x / z);
	} else if ((y <= 1.55e+104) || !(y <= 8.8e+256)) {
		tmp = y / (-z / x);
	} else {
		tmp = (y * x) / x;
	}
	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 <= 8.6d+43) then
        tmp = y + (x / z)
    else if ((y <= 1.55d+104) .or. (.not. (y <= 8.8d+256))) then
        tmp = y / (-z / x)
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e+43) {
		tmp = y + (x / z);
	} else if ((y <= 1.55e+104) || !(y <= 8.8e+256)) {
		tmp = y / (-z / x);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.6e+43:
		tmp = y + (x / z)
	elif (y <= 1.55e+104) or not (y <= 8.8e+256):
		tmp = y / (-z / x)
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.6e+43)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 1.55e+104) || !(y <= 8.8e+256))
		tmp = Float64(y / Float64(Float64(-z) / x));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.6e+43)
		tmp = y + (x / z);
	elseif ((y <= 1.55e+104) || ~((y <= 8.8e+256)))
		tmp = y / (-z / x);
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.6e+43], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.55e+104], N[Not[LessEqual[y, 8.8e+256]], $MachinePrecision]], N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.6e43

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 98.0%

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

   4. Taylor expanded in x around 0 89.0%

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

   if 8.6e43 < y < 1.55000000000000008e104 or 8.7999999999999997e256 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf 85.5%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. distribute-frac-neg2 79.3%

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

   8. Simplified 79.3%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-unprod 16.1%

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

      3. sqr-neg 16.1%

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

      4. sqrt-unprod 0.3%

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

      5. add-sqr-sqrt 1.3%

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

      6. clear-num 1.3%

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

      7. div-inv 1.3%

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

      8. frac-2neg 1.3%

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

      9. distribute-frac-neg 1.3%

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

      10. add-sqr-sqrt 1.1%

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

      11. sqrt-unprod 64.4%

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

      12. sqr-neg 64.4%

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

      13. sqrt-unprod 63.2%

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

      14. add-sqr-sqrt 79.3%

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

   10. Applied egg-rr 79.3%

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

   if 1.55000000000000008e104 < y < 8.7999999999999997e256

   1. Initial program 77.0%

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

   3. Taylor expanded in x around inf 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. mul-1-neg 64.0%

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

      4. unsub-neg 64.0%

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

      5. div-sub 64.0%

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

   5. Simplified 64.0%

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

   6. Taylor expanded in z around inf 22.8%

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

      1. associate-*r/ 63.7%

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

   8. Applied egg-rr 63.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+43}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+104} \lor \neg \left(y \leq 8.8 \cdot 10^{+256}\right):\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+89} \lor \neg \left(x \leq 9.6 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4e+89) (not (<= x 9.6e-16)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e+89) || !(x <= 9.6e-16)) {
		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 <= (-1.4d+89)) .or. (.not. (x <= 9.6d-16))) 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 <= -1.4e+89) || !(x <= 9.6e-16)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4e+89) or not (x <= 9.6e-16):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4e+89) || !(x <= 9.6e-16))
		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 <= -1.4e+89) || ~((x <= 9.6e-16)))
		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, -1.4e+89], N[Not[LessEqual[x, 9.6e-16]], $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 < -1.3999999999999999e89 or 9.60000000000000019e-16 < x

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. associate-/l* 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

   5. Simplified 88.0%

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

   if -1.3999999999999999e89 < x < 9.60000000000000019e-16

   1. Initial program 89.1%

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

   3. Taylor expanded in y around 0 99.2%

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

   4. Taylor expanded in x around 0 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+89} \lor \neg \left(x \leq 9.6 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -380000.0) (not (<= y 1.0))) (* y (/ (- z x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380000.0) || !(y <= 1.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-380000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * ((z - x) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380000.0) || !(y <= 1.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -380000.0) or not (y <= 1.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -380000.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -380000.0) || ~((y <= 1.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -380000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e5 or 1 < y

   1. Initial program 80.8%

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

   3. Taylor expanded in y around inf 80.6%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -3.8e5 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 96.9%

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

   4. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -380000.0)
   (* y (/ (- z x) z))
   (if (<= y 1.0) (+ y (/ x z)) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -380000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-380000.0d0)) then
        tmp = y * ((z - x) / z)
    else if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -380000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -380000.0:
		tmp = y * ((z - x) / z)
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -380000.0)
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -380000.0)
		tmp = y * ((z - x) / z);
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -380000.0], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e5

   1. Initial program 79.4%

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

   3. Taylor expanded in y around inf 79.1%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   if -3.8e5 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 96.9%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 1 < y

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0 91.1%

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

   4. Taylor expanded in x around 0 91.6%

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

      1. neg-mul-1 91.6%

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

      2. +-commutative 91.6%

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

      3. sub-neg 91.6%

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

      4. div-sub 91.6%

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

      5. associate-/l* 93.0%

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

      6. associate-*l/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. associate-*l/ 99.9%

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

      3. distribute-rgt-neg-out 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e+122) (+ y (/ x z)) (/ (* y x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+122) {
		tmp = y + (x / z);
	} else {
		tmp = (y * x) / x;
	}
	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+122) then
        tmp = y + (x / z)
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+122) {
		tmp = y + (x / z);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.1e+122:
		tmp = y + (x / z)
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e+122)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.1e+122)
		tmp = y + (x / z);
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+122], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000016e122

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0 97.6%

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

   4. Taylor expanded in x around 0 85.0%

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

   if 2.10000000000000016e122 < y

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 63.5%

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

      1. associate-+r+ 63.5%

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

      2. +-commutative 63.5%

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

      3. mul-1-neg 63.5%

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

      4. unsub-neg 63.5%

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

      5. div-sub 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in z around inf 16.6%

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

      1. associate-*r/ 55.5%

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

   8. Applied egg-rr 55.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ y + \frac{x}{z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in y around 0 96.0%

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

4. Taylor expanded in x around 0 78.6%

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

5. Final simplification 78.6%

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

Alternative 11: 40.0% 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 90.4%

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

3. Taylor expanded in x around 0 42.6%

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

4. Final simplification 42.6%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 94.5% 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 2024081 
(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))