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

Percentage Accurate: 88.3% → 99.8%

Time: 4.8s

Alternatives: 9

Speedup: 0.8×

• 21.2% 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: 88.3% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+47} \lor \neg \left(z \leq 10^{-7}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.5e+47) (not (<= z 1e-7)))
   (+ (/ x z) (* (- 1.0 (/ x z)) y))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e+47) || !(z <= 1e-7)) {
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8.5d+47)) .or. (.not. (z <= 1d-7))) then
        tmp = (x / z) + ((1.0d0 - (x / z)) * y)
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e+47) || !(z <= 1e-7)) {
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.5e+47) or not (z <= 1e-7):
		tmp = (x / z) + ((1.0 - (x / z)) * y)
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.5e+47) || !(z <= 1e-7))
		tmp = Float64(Float64(x / z) + Float64(Float64(1.0 - Float64(x / z)) * y));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.5e+47) || ~((z <= 1e-7)))
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.5e+47], N[Not[LessEqual[z, 1e-7]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] + N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000008e47 or 9.9999999999999995e-8 < z

   1. Initial program 78.8%

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

   2. Taylor expanded in y around 0 100.0%

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

   if -8.5000000000000008e47 < z < 9.9999999999999995e-8

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+47} \lor \neg \left(z \leq 10^{-7}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000000000000 \lor \neg \left(y \leq 4 \cdot 10^{+49}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -10000000000000.0) (not (<= y 4e+49)))
   (* (- 1.0 (/ x z)) y)
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -10000000000000.0) || !(y <= 4e+49)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-10000000000000.0d0)) .or. (.not. (y <= 4d+49))) then
        tmp = (1.0d0 - (x / z)) * y
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -10000000000000.0) || !(y <= 4e+49)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -10000000000000.0) or not (y <= 4e+49):
		tmp = (1.0 - (x / z)) * y
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -10000000000000.0) || !(y <= 4e+49))
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -10000000000000.0) || ~((y <= 4e+49)))
		tmp = (1.0 - (x / z)) * y;
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -10000000000000.0], N[Not[LessEqual[y, 4e+49]], $MachinePrecision]], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e13 or 3.99999999999999979e49 < y

   1. Initial program 79.9%

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

   2. Taylor expanded in y around 0 89.3%

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

   3. Taylor expanded in y around inf 99.9%

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

   if -1e13 < y < 3.99999999999999979e49

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000000000 \lor \neg \left(y \leq 4 \cdot 10^{+49}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1600000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1600000.0) (not (<= y 1.0)))
   (* (- 1.0 (/ x z)) y)
   (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1600000.0) || !(y <= 1.0)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x / z) + 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 <= (-1600000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (1.0d0 - (x / z)) * y
    else
        tmp = (x / z) + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1600000.0) || !(y <= 1.0)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1600000.0) or not (y <= 1.0):
		tmp = (1.0 - (x / z)) * y
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1600000.0) || !(y <= 1.0))
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	else
		tmp = Float64(Float64(x / z) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1600000.0) || ~((y <= 1.0)))
		tmp = (1.0 - (x / z)) * y;
	else
		tmp = (x / z) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1600000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e6 or 1 < y

   1. Initial program 82.0%

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

   2. Taylor expanded in y around 0 85.6%

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

   3. Taylor expanded in y around inf 99.4%

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

   if -1.6e6 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 93.1%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1600000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 4: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-56} \lor \neg \left(y \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e-56) (not (<= y 1.35e-6))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e-56) || !(y <= 1.35e-6)) {
		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 <= (-1.05d-56)) .or. (.not. (y <= 1.35d-6))) 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 <= -1.05e-56) || !(y <= 1.35e-6)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.05e-56) or not (y <= 1.35e-6):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e-56) || !(y <= 1.35e-6))
		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 <= -1.05e-56) || ~((y <= 1.35e-6)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e-56], N[Not[LessEqual[y, 1.35e-6]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000003e-56 or 1.34999999999999999e-6 < y

   1. Initial program 83.8%

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

   2. Taylor expanded in y around inf 80.5%

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

   3. Taylor expanded in z around inf 34.0%

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

      1. associate-/l* 45.0%

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

      2. associate-/r/ 52.3%

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

   5. Applied egg-rr 52.3%

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

   if -1.05000000000000003e-56 < y < 1.34999999999999999e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-56} \lor \neg \left(y \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05e+30)
   (* (/ x z) (- y))
   (if (<= y 1.0) (+ (/ x z) y) (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05e+30) {
		tmp = (x / z) * -y;
	} else if (y <= 1.0) {
		tmp = (x / z) + y;
	} 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 <= (-1.05d+30)) then
        tmp = (x / z) * -y
    else if (y <= 1.0d0) then
        tmp = (x / z) + y
    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 <= -1.05e+30) {
		tmp = (x / z) * -y;
	} else if (y <= 1.0) {
		tmp = (x / z) + y;
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.05e+30:
		tmp = (x / z) * -y
	elif y <= 1.0:
		tmp = (x / z) + y
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05e+30)
		tmp = Float64(Float64(x / z) * Float64(-y));
	elseif (y <= 1.0)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.05e+30)
		tmp = (x / z) * -y;
	elseif (y <= 1.0)
		tmp = (x / z) + y;
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05e+30], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e30

   1. Initial program 90.8%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in y around inf 99.9%

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

   4. Taylor expanded in x around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. distribute-frac-neg 64.5%

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

   6. Simplified 64.5%

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

   if -1.05e30 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 93.2%

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

   3. Taylor expanded in x around 0 98.9%

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

   4. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

   6. Simplified 98.9%

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

   if 1 < y

   1. Initial program 74.8%

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

   2. Taylor expanded in y around 0 74.5%

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

   3. Taylor expanded in x around 0 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 31.1%

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

      4. sqrt-unprod 58.1%

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

      5. sqr-neg 58.1%

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

      6. sqrt-unprod 27.7%

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

      7. add-sqr-sqrt 70.6%

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

      8. cancel-sign-sub-inv 70.6%

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

      9. div-inv 70.6%

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

   5. Applied egg-rr 70.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]

Alternative 6: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.6e-55) y (if (<= y 0.49) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.6e-55) {
		tmp = y;
	} else if (y <= 0.49) {
		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 <= (-5.6d-55)) then
        tmp = y
    else if (y <= 0.49d0) 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 <= -5.6e-55) {
		tmp = y;
	} else if (y <= 0.49) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.6e-55:
		tmp = y
	elif y <= 0.49:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.6e-55)
		tmp = y;
	elseif (y <= 0.49)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.6e-55)
		tmp = y;
	elseif (y <= 0.49)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.6e-55], y, If[LessEqual[y, 0.49], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -5.59999999999999968e-55 or 0.48999999999999999 < y

   1. Initial program 83.8%

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

   2. Taylor expanded in x around 0 45.0%

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

   if -5.59999999999999968e-55 < y < 0.48999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.4%

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

   2. Taylor expanded in y around 0 95.1%

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

   3. Taylor expanded in x around 0 84.1%

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

   4. Taylor expanded in y around 0 84.1%

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

      1. +-commutative 84.1%

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

   6. Simplified 84.1%

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

   if 1 < y

   1. Initial program 74.8%

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

   2. Taylor expanded in y around inf 74.4%

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

   3. Taylor expanded in z around inf 28.2%

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

      1. associate-/l* 45.7%

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

      2. associate-/r/ 55.0%

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

   5. Applied egg-rr 55.0%

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

4. Final simplification 76.1%

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

Alternative 8: 81.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.4%

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

   2. Taylor expanded in y around 0 95.1%

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

   3. Taylor expanded in x around 0 84.1%

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

   4. Taylor expanded in y around 0 84.1%

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

      1. +-commutative 84.1%

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

   6. Simplified 84.1%

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

   if 1 < y

   1. Initial program 74.8%

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

   2. Taylor expanded in y around 0 74.5%

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

   3. Taylor expanded in x around 0 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 31.1%

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

      4. sqrt-unprod 58.1%

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

      5. sqr-neg 58.1%

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

      6. sqrt-unprod 27.7%

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

      7. add-sqr-sqrt 70.6%

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

      8. cancel-sign-sub-inv 70.6%

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

      9. div-inv 70.6%

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

   5. Applied egg-rr 70.6%

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

4. Final simplification 80.4%

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

Alternative 9: 40.5% 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 91.1%

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

2. Taylor expanded in x around 0 33.0%

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

3. Final simplification 33.0%

   \[\leadsto y \]

Developer target: 93.7% 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 2023200 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

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

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