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

Percentage Accurate: 88.1% → 97.7%

Time: 6.6s

Alternatives: 9

Speedup: 0.6×

• 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: 88.1% 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: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e+51) (* y (- 1.0 (/ x z))) (+ y (/ (* x (- 1.0 y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+51) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x * (1.0 - 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 <= (-5d+51)) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + ((x * (1.0d0 - y)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+51) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x * (1.0 - y)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e+51:
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + ((x * (1.0 - y)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e+51)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(Float64(x * Float64(1.0 - y)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e+51)
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + ((x * (1.0 - y)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e+51], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e51

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf 75.3%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   if -5e51 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf 91.7%

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

      1. associate-+r+ 91.7%

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

      2. +-commutative 91.7%

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

      3. mul-1-neg 91.7%

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

      4. unsub-neg 91.7%

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

      5. div-sub 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in z around inf 99.0%

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

Alternative 2: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ t_1 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))) (t_1 (* y (/ (- x) z))))
   (if (<= y -4.1e+121)
     t_0
     (if (<= y -2.5e+37)
       t_1
       (if (<= y 6.4e+19) t_0 (if (<= y 1.52e+75) t_1 (/ (* y x) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double t_1 = y * (-x / z);
	double tmp;
	if (y <= -4.1e+121) {
		tmp = t_0;
	} else if (y <= -2.5e+37) {
		tmp = t_1;
	} else if (y <= 6.4e+19) {
		tmp = t_0;
	} else if (y <= 1.52e+75) {
		tmp = t_1;
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x / z)
    t_1 = y * (-x / z)
    if (y <= (-4.1d+121)) then
        tmp = t_0
    else if (y <= (-2.5d+37)) then
        tmp = t_1
    else if (y <= 6.4d+19) then
        tmp = t_0
    else if (y <= 1.52d+75) then
        tmp = t_1
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double t_1 = y * (-x / z);
	double tmp;
	if (y <= -4.1e+121) {
		tmp = t_0;
	} else if (y <= -2.5e+37) {
		tmp = t_1;
	} else if (y <= 6.4e+19) {
		tmp = t_0;
	} else if (y <= 1.52e+75) {
		tmp = t_1;
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	t_1 = y * (-x / z)
	tmp = 0
	if y <= -4.1e+121:
		tmp = t_0
	elif y <= -2.5e+37:
		tmp = t_1
	elif y <= 6.4e+19:
		tmp = t_0
	elif y <= 1.52e+75:
		tmp = t_1
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	t_1 = Float64(y * Float64(Float64(-x) / z))
	tmp = 0.0
	if (y <= -4.1e+121)
		tmp = t_0;
	elseif (y <= -2.5e+37)
		tmp = t_1;
	elseif (y <= 6.4e+19)
		tmp = t_0;
	elseif (y <= 1.52e+75)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	t_1 = y * (-x / z);
	tmp = 0.0;
	if (y <= -4.1e+121)
		tmp = t_0;
	elseif (y <= -2.5e+37)
		tmp = t_1;
	elseif (y <= 6.4e+19)
		tmp = t_0;
	elseif (y <= 1.52e+75)
		tmp = t_1;
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.1e+121], t$95$0, If[LessEqual[y, -2.5e+37], t$95$1, If[LessEqual[y, 6.4e+19], t$95$0, If[LessEqual[y, 1.52e+75], t$95$1, N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e121 or -2.49999999999999994e37 < y < 6.4e19

   1. Initial program 92.5%

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

   3. Taylor expanded in z around inf 85.7%

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

   4. Taylor expanded in x around 0 91.7%

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

      1. +-commutative 91.7%

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

   6. Simplified 91.7%

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

   if -4.1e121 < y < -2.49999999999999994e37 or 6.4e19 < y < 1.5199999999999999e75

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. mul-1-neg 82.5%

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

      3. distribute-rgt-neg-out 82.5%

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

   8. Simplified 82.5%

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

   9. Taylor expanded in x around 0 82.5%

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

       1. associate-*r/ 82.5%

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

       2. *-commutative 82.5%

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

       3. mul-1-neg 82.5%

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

       4. distribute-rgt-neg-in 82.5%

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

       5. associate-*r/ 82.6%

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

   11. Simplified 82.6%

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

   if 1.5199999999999999e75 < y

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 61.4%

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

      1. associate-+r+ 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. unsub-neg 61.4%

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

      5. div-sub 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in z around inf 23.0%

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

      1. *-commutative 23.0%

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

      2. associate-*l/ 66.1%

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

   8. Applied egg-rr 66.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+121}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+75}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < 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 81.5%

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

      1. associate-/l* 98.5%

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

      2. div-sub 98.5%

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

      3. *-inverses 98.5%

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

   5. Simplified 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.1%

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

Alternative 4: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 79.9%

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

   3. Taylor expanded in y around inf 77.3%

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

      1. associate-/l* 97.3%

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

      2. div-sub 97.4%

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

      3. *-inverses 97.4%

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

   5. Simplified 97.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 1 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf 87.1%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.1%

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

Alternative 5: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e+22)
   (+ y (/ x z))
   (if (<= y 4.8e+67) (* x (/ y (- z))) (/ (* y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+22) {
		tmp = y + (x / z);
	} else if (y <= 4.8e+67) {
		tmp = x * (y / -z);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.2d+22) then
        tmp = y + (x / z)
    else if (y <= 4.8d+67) then
        tmp = x * (y / -z)
    else
        tmp = (y * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+22) {
		tmp = y + (x / z);
	} else if (y <= 4.8e+67) {
		tmp = x * (y / -z);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e+22:
		tmp = y + (x / z)
	elif y <= 4.8e+67:
		tmp = x * (y / -z)
	else:
		tmp = (y * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e+22)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 4.8e+67)
		tmp = Float64(x * Float64(y / Float64(-z)));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e+22)
		tmp = y + (x / z);
	elseif (y <= 4.8e+67)
		tmp = x * (y / -z);
	else
		tmp = (y * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e+22], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+67], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2000000000000008e22

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf 82.3%

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

   4. Taylor expanded in x around 0 87.8%

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

      1. +-commutative 87.8%

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

   6. Simplified 87.8%

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

   if 9.2000000000000008e22 < y < 4.80000000000000004e67

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.8%

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

      3. *-inverses 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. distribute-neg-frac 99.7%

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

   8. Simplified 99.7%

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

   if 4.80000000000000004e67 < y

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 61.4%

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

      1. associate-+r+ 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. unsub-neg 61.4%

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

      5. div-sub 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in z around inf 23.0%

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

      1. *-commutative 23.0%

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

      2. associate-*l/ 66.1%

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

   8. Applied egg-rr 66.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+22}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -30.0) y (if (<= y 2.3e-37) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -30.0) {
		tmp = y;
	} else if (y <= 2.3e-37) {
		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 <= (-30.0d0)) then
        tmp = y
    else if (y <= 2.3d-37) 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 <= -30.0) {
		tmp = y;
	} else if (y <= 2.3e-37) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -30.0:
		tmp = y
	elif y <= 2.3e-37:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -30.0)
		tmp = y;
	elseif (y <= 2.3e-37)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -30.0)
		tmp = y;
	elseif (y <= 2.3e-37)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -30.0], y, If[LessEqual[y, 2.3e-37], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -30 or 2.3e-37 < y

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0 49.3%

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

   if -30 < y < 2.3e-37

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 70.7%

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

Alternative 7: 78.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 93.1%

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

   3. Taylor expanded in z around inf 82.2%

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

   4. Taylor expanded in x around 0 87.7%

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

      1. +-commutative 87.7%

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

   6. Simplified 87.7%

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

   if 1 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in x around inf 70.4%

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

      1. associate-+r+ 70.4%

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

      2. +-commutative 70.4%

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

      3. mul-1-neg 70.4%

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

      4. unsub-neg 70.4%

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

      5. div-sub 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in z around inf 21.9%

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

      1. *-commutative 21.9%

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

      2. associate-*l/ 54.9%

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

   8. Applied egg-rr 54.9%

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

4. Final simplification 81.2%

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

Alternative 8: 78.0% 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 91.9%

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

3. Taylor expanded in z around inf 72.9%

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

4. Taylor expanded in x around 0 79.5%

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

   1. +-commutative 79.5%

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

6. Simplified 79.5%

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

7. Final simplification 79.5%

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

Alternative 9: 41.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	return y
end

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 91.9%

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

3. Taylor expanded in x around 0 40.0%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Developer target: 94.1% 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 2024090 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (- (+ y (/ x z)) (/ y (/ z x)))

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