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

Percentage Accurate: 87.7% → 98.4%

Time: 6.2s

Alternatives: 10

Speedup: 0.8×

• 20.8% 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: 87.7% 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq 4 \cdot 10^{-188}:\\ \;\;\;\;\frac{x}{z} + t_0\\ \mathbf{elif}\;y \leq 200000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (/ x z)))))
   (if (<= y 4e-188)
     (+ (/ x z) t_0)
     (if (<= y 200000.0) (/ (+ x (* y (- z x))) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= 4e-188) {
		tmp = (x / z) + t_0;
	} else if (y <= 200000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - (x / z))
    if (y <= 4d-188) then
        tmp = (x / z) + t_0
    else if (y <= 200000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= 4e-188) {
		tmp = (x / z) + t_0;
	} else if (y <= 200000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - (x / z))
	tmp = 0
	if y <= 4e-188:
		tmp = (x / z) + t_0
	elif y <= 200000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - Float64(x / z)))
	tmp = 0.0
	if (y <= 4e-188)
		tmp = Float64(Float64(x / z) + t_0);
	elseif (y <= 200000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - (x / z));
	tmp = 0.0;
	if (y <= 4e-188)
		tmp = (x / z) + t_0;
	elseif (y <= 200000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4e-188], N[(N[(x / z), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[y, 200000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.9999999999999998e-188

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 3.9999999999999998e-188 < y < 2e5

   1. Initial program 100.0%

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

   if 2e5 < y

   1. Initial program 75.7%

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

   2. Taylor expanded in x around 0 96.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-188}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 200000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+38} \lor \neg \left(y \leq 200000\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+38) (not (<= y 200000.0)))
   (* y (- 1.0 (/ x z)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.5e+38) || !(y <= 200000.0)) {
		tmp = y * (1.0 - (x / z));
	} 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 <= (-3.5d+38)) .or. (.not. (y <= 200000.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    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 <= -3.5e+38) || !(y <= 200000.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.5e+38) or not (y <= 200000.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.5e+38) || !(y <= 200000.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	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 <= -3.5e+38) || ~((y <= 200000.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e+38], N[Not[LessEqual[y, 200000.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000002e38 or 2e5 < y

   1. Initial program 73.3%

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

   2. Taylor expanded in x around 0 95.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -3.50000000000000002e38 < y < 2e5

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+38} \lor \neg \left(y \leq 200000\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 3: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+244} \lor \neg \left(y \leq 2.3 \cdot 10^{+283}\right):\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.0)
   (+ y (/ x z))
   (if (or (<= y 5.1e+244) (not (<= y 2.3e+283)))
     (- y (/ x z))
     (* y (/ (- x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else if ((y <= 5.1e+244) || !(y <= 2.3e+283)) {
		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 <= 1.0d0) then
        tmp = y + (x / z)
    else if ((y <= 5.1d+244) .or. (.not. (y <= 2.3d+283))) 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 <= 1.0) {
		tmp = y + (x / z);
	} else if ((y <= 5.1e+244) || !(y <= 2.3e+283)) {
		tmp = y - (x / z);
	} else {
		tmp = y * (-x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	elif (y <= 5.1e+244) or not (y <= 2.3e+283):
		tmp = y - (x / z)
	else:
		tmp = y * (-x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 5.1e+244) || !(y <= 2.3e+283))
		tmp = Float64(y - Float64(x / z));
	else
		tmp = Float64(y * Float64(Float64(-x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = y + (x / z);
	elseif ((y <= 5.1e+244) || ~((y <= 2.3e+283)))
		tmp = y - (x / z);
	else
		tmp = y * (-x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 5.1e+244], N[Not[LessEqual[y, 2.3e+283]], $MachinePrecision]], 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

   1. Initial program 91.2%

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

   2. Taylor expanded in x around 0 98.3%

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

   3. Taylor expanded in y around 0 91.0%

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

   if 1 < y < 5.09999999999999985e244 or 2.3000000000000002e283 < y

   1. Initial program 73.7%

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

   2. Taylor expanded in x around 0 96.7%

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

   3. Taylor expanded in y around 0 45.2%

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

      1. frac-2neg 45.2%

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

      2. add-sqr-sqrt 26.8%

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

      3. sqrt-unprod 59.1%

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

      4. sqr-neg 59.1%

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

      5. sqrt-unprod 32.2%

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

      6. add-sqr-sqrt 74.4%

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

      7. distribute-neg-frac 74.4%

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

      8. sub-neg 74.4%

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

   5. Applied egg-rr 74.4%

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

   if 5.09999999999999985e244 < y < 2.3000000000000002e283

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. associate-/l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. un-div-inv 100.0%

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

      6. clear-num 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+244} \lor \neg \left(y \leq 2.3 \cdot 10^{+283}\right):\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+27} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.5e+27) (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 <= -3.5e+27) || !(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 <= (-3.5d+27)) .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 <= -3.5e+27) || !(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 <= -3.5e+27) 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 <= -3.5e+27) || !(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 <= -3.5e+27) || ~((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, -3.5e+27], 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 < -3.5000000000000002e27 or 1 < y

   1. Initial program 73.7%

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

   2. Taylor expanded in x around 0 96.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   if -3.5000000000000002e27 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around 0 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+27} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5: 96.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

2. Taylor expanded in x around 0 98.0%

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

3. Final simplification 98.0%

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

Alternative 6: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+69} \lor \neg \left(y \leq 1.5 \cdot 10^{-15}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.8e+69) (not (<= y 1.5e-15))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.8e+69) || !(y <= 1.5e-15)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4.8d+69)) .or. (.not. (y <= 1.5d-15))) then
        tmp = z * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.8e+69) || !(y <= 1.5e-15)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.8e+69) or not (y <= 1.5e-15):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.8e+69) || !(y <= 1.5e-15))
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.8e+69) || ~((y <= 1.5e-15)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.8e+69], N[Not[LessEqual[y, 1.5e-15]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e69 or 1.5e-15 < y

   1. Initial program 73.4%

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

   2. Taylor expanded in x around 0 28.9%

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

      1. associate-/l* 50.5%

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

      2. associate-/r/ 55.2%

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

   4. Applied egg-rr 55.2%

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

   if -4.8000000000000003e69 < y < 1.5e-15

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 73.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+69} \lor \neg \left(y \leq 1.5 \cdot 10^{-15}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+20}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+79}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e+20)
   (+ y (/ x z))
   (if (<= y 2.45e+79) (/ (- x) z) (* z (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e+20) {
		tmp = y + (x / z);
	} else if (y <= 2.45e+79) {
		tmp = -x / z;
	} 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 <= 7.2d+20) then
        tmp = y + (x / z)
    else if (y <= 2.45d+79) then
        tmp = -x / z
    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 <= 7.2e+20) {
		tmp = y + (x / z);
	} else if (y <= 2.45e+79) {
		tmp = -x / z;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.2e+20:
		tmp = y + (x / z)
	elif y <= 2.45e+79:
		tmp = -x / z
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.2e+20)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 2.45e+79)
		tmp = Float64(Float64(-x) / z);
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.2e+20)
		tmp = y + (x / z);
	elseif (y <= 2.45e+79)
		tmp = -x / z;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.2e+20], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e+79], N[((-x) / z), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2e20

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 98.4%

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

   3. Taylor expanded in y around 0 90.1%

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

   if 7.2e20 < y < 2.4499999999999999e79

   1. Initial program 86.2%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around 0 7.5%

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

      1. frac-2neg 7.5%

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

      2. add-sqr-sqrt 0.3%

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

      3. sqrt-unprod 36.7%

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

      4. sqr-neg 36.7%

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

      5. sqrt-unprod 29.7%

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

      6. add-sqr-sqrt 53.3%

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

      7. distribute-neg-frac 53.3%

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

      8. sub-neg 53.3%

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

   5. Applied egg-rr 53.3%

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

   6. Taylor expanded in y around 0 47.0%

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

      1. mul-1-neg 47.0%

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

      2. distribute-frac-neg 47.0%

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

   8. Simplified 47.0%

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

   if 2.4499999999999999e79 < y

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0 24.2%

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

      1. associate-/l* 50.5%

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

      2. associate-/r/ 59.0%

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

   4. Applied egg-rr 59.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+20}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+79}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 8: 60.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e+68) y (if (<= y 3.3e-15) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e+68) {
		tmp = y;
	} else if (y <= 3.3e-15) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.05d+68)) then
        tmp = y
    else if (y <= 3.3d-15) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e+68) {
		tmp = y;
	} else if (y <= 3.3e-15) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.05e+68:
		tmp = y
	elif y <= 3.3e-15:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.05e+68)
		tmp = y;
	elseif (y <= 3.3e-15)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.05e+68)
		tmp = y;
	elseif (y <= 3.3e-15)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.05e+68], y, If[LessEqual[y, 3.3e-15], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05e68 or 3.3e-15 < y

   1. Initial program 73.4%

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

   2. Taylor expanded in x around 0 50.5%

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

   if -2.05e68 < y < 3.3e-15

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 73.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9: 82.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z):
	tmp = 0
	if 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 <= 1.0)
		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 <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 91.2%

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

   2. Taylor expanded in x around 0 98.3%

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

   3. Taylor expanded in y around 0 91.0%

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

   if 1 < y

   1. Initial program 75.7%

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

   2. Taylor expanded in x around 0 96.9%

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

   3. Taylor expanded in y around 0 41.9%

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

      1. frac-2neg 41.9%

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

      2. add-sqr-sqrt 24.8%

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

      3. sqrt-unprod 56.2%

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

      4. sqr-neg 56.2%

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

      5. sqrt-unprod 29.8%

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

      6. add-sqr-sqrt 68.9%

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

      7. distribute-neg-frac 68.9%

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

      8. sub-neg 68.9%

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

   5. Applied egg-rr 68.9%

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

4. Final simplification 85.4%

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

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 87.2%

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

2. Taylor expanded in x around 0 37.1%

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

3. Final simplification 37.1%

   \[\leadsto y \]

Developer target: 93.5% 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 2023318 
(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))