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

Percentage Accurate: 87.2% → 99.9%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 20.3% 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: 87.2% 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

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

FPCore

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

C

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

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 + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

3. Taylor expanded in x around -inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. associate-/l* 98.1%

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

   4. associate-/r/ 100.0%

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

   5. sub-neg 100.0%

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

   6. metadata-eval 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-10} \lor \neg \left(x \leq 1.5 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e-10) (not (<= x 1.5e+124)))
   (* (/ x z) (- 1.0 y))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e-10) || !(x <= 1.5e+124)) {
		tmp = (x / z) * (1.0 - 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 ((x <= (-8.5d-10)) .or. (.not. (x <= 1.5d+124))) then
        tmp = (x / z) * (1.0d0 - 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 ((x <= -8.5e-10) || !(x <= 1.5e+124)) {
		tmp = (x / z) * (1.0 - y);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.5e-10) or not (x <= 1.5e+124):
		tmp = (x / z) * (1.0 - y)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e-10) || !(x <= 1.5e+124))
		tmp = Float64(Float64(x / z) * Float64(1.0 - y));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.5e-10) || ~((x <= 1.5e+124)))
		tmp = (x / z) * (1.0 - y);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.5e-10], N[Not[LessEqual[x, 1.5e+124]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999996e-10 or 1.5e124 < x

   1. Initial program 89.3%

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

   3. Taylor expanded in x around inf 83.8%

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

      1. associate-/l* 89.5%

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

      2. associate-/r/ 89.5%

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

      3. mul-1-neg 89.5%

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

      4. unsub-neg 89.5%

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

   5. Simplified 89.5%

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

   if -8.4999999999999996e-10 < x < 1.5e124

   1. Initial program 88.7%

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

   3. Taylor expanded in x around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. associate-/l* 96.6%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-frac-neg 88.5%

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

   8. Simplified 88.5%

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

      1. div-inv 88.4%

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

      2. cancel-sign-sub 88.4%

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

      3. div-inv 88.5%

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

      4. +-commutative 88.5%

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

   10. Applied egg-rr 88.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-10} \lor \neg \left(x \leq 1.5 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 78.7%

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

   3. Taylor expanded in y around inf 78.2%

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

      1. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

   8. Simplified 98.2%

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

      1. div-inv 97.9%

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

      2. cancel-sign-sub 97.9%

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

      3. div-inv 98.2%

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

      4. +-commutative 98.2%

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

   10. Applied egg-rr 98.2%

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

4. Final simplification 98.8%

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

Alternative 4: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -116:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -116.0) y (if (<= y 2.6e-64) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -116.0) {
		tmp = y;
	} else if (y <= 2.6e-64) {
		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 <= (-116.0d0)) then
        tmp = y
    else if (y <= 2.6d-64) 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 <= -116.0) {
		tmp = y;
	} else if (y <= 2.6e-64) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -116.0:
		tmp = y
	elif y <= 2.6e-64:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -116.0)
		tmp = y;
	elseif (y <= 2.6e-64)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -116.0)
		tmp = y;
	elseif (y <= 2.6e-64)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -116.0], y, If[LessEqual[y, 2.6e-64], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -116 or 2.6e-64 < y

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 52.4%

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

   if -116 < y < 2.6e-64

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -116:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+62) (+ y (/ x z)) (* y (- (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+62) {
		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 <= 1.9d+62) 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 <= 1.9e+62) {
		tmp = y + (x / z);
	} else {
		tmp = y * -(x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.9e+62:
		tmp = y + (x / z)
	else:
		tmp = y * -(x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+62)
		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 <= 1.9e+62)
		tmp = y + (x / z);
	else
		tmp = y * -(x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e+62], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * (-N[(x / z), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999992e62

   1. Initial program 90.4%

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

   3. Taylor expanded in x around -inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. unsub-neg 96.8%

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

      3. associate-/l* 98.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. distribute-frac-neg 86.0%

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

   8. Simplified 86.0%

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

      1. div-inv 85.8%

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

      2. cancel-sign-sub 85.8%

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

      3. div-inv 86.0%

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

      4. +-commutative 86.0%

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

   10. Applied egg-rr 86.0%

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

   if 1.89999999999999992e62 < y

   1. Initial program 82.9%

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

   3. Taylor expanded in x around inf 60.1%

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

      1. associate-/l* 62.9%

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

      2. associate-/r/ 64.0%

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

      3. mul-1-neg 64.0%

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

      4. unsub-neg 64.0%

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

   5. Simplified 64.0%

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

   6. Taylor expanded in y around inf 60.1%

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

      1. mul-1-neg 60.1%

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

      2. associate-*l/ 64.0%

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

      3. distribute-rgt-neg-out 64.0%

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

   8. Simplified 64.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+62}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+62) (+ y (/ x z)) (/ y (/ (- z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+62) {
		tmp = y + (x / z);
	} else {
		tmp = y / (-z / 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 <= 8.2d+62) then
        tmp = y + (x / z)
    else
        tmp = y / (-z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+62) {
		tmp = y + (x / z);
	} else {
		tmp = y / (-z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e+62:
		tmp = y + (x / z)
	else:
		tmp = y / (-z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+62)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y / Float64(Float64(-z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e+62)
		tmp = y + (x / z);
	else
		tmp = y / (-z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e+62], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.19999999999999967e62

   1. Initial program 90.4%

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

   3. Taylor expanded in x around -inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. unsub-neg 96.8%

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

      3. associate-/l* 98.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. distribute-frac-neg 86.0%

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

   8. Simplified 86.0%

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

      1. div-inv 85.8%

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

      2. cancel-sign-sub 85.8%

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

      3. div-inv 86.0%

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

      4. +-commutative 86.0%

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

   10. Applied egg-rr 86.0%

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

   if 8.19999999999999967e62 < y

   1. Initial program 82.9%

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

   3. Taylor expanded in y around inf 82.9%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. neg-mul-1 64.1%

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

   8. Simplified 64.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+62}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% 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 89.0%

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

3. Taylor expanded in x around -inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. associate-/l* 98.1%

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

   4. associate-/r/ 100.0%

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

   5. sub-neg 100.0%

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

   6. metadata-eval 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in y around 0 76.5%

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

   1. mul-1-neg 76.5%

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

   2. distribute-frac-neg 76.5%

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

8. Simplified 76.5%

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

   1. div-inv 76.4%

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

   2. cancel-sign-sub 76.4%

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

   3. div-inv 76.5%

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

   4. +-commutative 76.5%

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

10. Applied egg-rr 76.5%

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

11. Final simplification 76.5%

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

Alternative 8: 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 89.0%

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

3. Taylor expanded in x around 0 37.7%

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

4. Final simplification 37.7%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 94.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(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))