Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 8

Speedup: 1.0×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / 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 = 1.0d0 + ((4.0d0 * ((x + (y * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / 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 = 1.0d0 + ((4.0d0 * ((x + (y * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / 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 = 1.0d0 + ((4.0d0 * ((x + (y * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
4. Add Preprocessing

Alternative 2: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z \cdot -4}{y}\\ \mathbf{if}\;y \leq -1.48 \cdot 10^{+53}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-285}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* z -4.0) y))))
   (if (<= y -1.48e+53)
     2.0
     (if (<= y -9.5e-56)
       t_0
       (if (<= y 8.5e-285)
         (+ 1.0 (/ (* 4.0 x) y))
         (if (<= y 3.9e+79) t_0 2.0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z * -4.0) / y);
	double tmp;
	if (y <= -1.48e+53) {
		tmp = 2.0;
	} else if (y <= -9.5e-56) {
		tmp = t_0;
	} else if (y <= 8.5e-285) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else if (y <= 3.9e+79) {
		tmp = t_0;
	} else {
		tmp = 2.0;
	}
	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 = 1.0d0 + ((z * (-4.0d0)) / y)
    if (y <= (-1.48d+53)) then
        tmp = 2.0d0
    else if (y <= (-9.5d-56)) then
        tmp = t_0
    else if (y <= 8.5d-285) then
        tmp = 1.0d0 + ((4.0d0 * x) / y)
    else if (y <= 3.9d+79) then
        tmp = t_0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z * -4.0) / y);
	double tmp;
	if (y <= -1.48e+53) {
		tmp = 2.0;
	} else if (y <= -9.5e-56) {
		tmp = t_0;
	} else if (y <= 8.5e-285) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else if (y <= 3.9e+79) {
		tmp = t_0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((z * -4.0) / y)
	tmp = 0
	if y <= -1.48e+53:
		tmp = 2.0
	elif y <= -9.5e-56:
		tmp = t_0
	elif y <= 8.5e-285:
		tmp = 1.0 + ((4.0 * x) / y)
	elif y <= 3.9e+79:
		tmp = t_0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(z * -4.0) / y))
	tmp = 0.0
	if (y <= -1.48e+53)
		tmp = 2.0;
	elseif (y <= -9.5e-56)
		tmp = t_0;
	elseif (y <= 8.5e-285)
		tmp = Float64(1.0 + Float64(Float64(4.0 * x) / y));
	elseif (y <= 3.9e+79)
		tmp = t_0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((z * -4.0) / y);
	tmp = 0.0;
	if (y <= -1.48e+53)
		tmp = 2.0;
	elseif (y <= -9.5e-56)
		tmp = t_0;
	elseif (y <= 8.5e-285)
		tmp = 1.0 + ((4.0 * x) / y);
	elseif (y <= 3.9e+79)
		tmp = t_0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.48e+53], 2.0, If[LessEqual[y, -9.5e-56], t$95$0, If[LessEqual[y, 8.5e-285], N[(1.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+79], t$95$0, 2.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.48e53 or 3.8999999999999997e79 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 74.9%

      \[\leadsto \color{blue}{2} \]

   if -1.48e53 < y < -9.4999999999999991e-56 or 8.49999999999999979e-285 < y < 3.8999999999999997e79

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 58.3%

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

      1. *-commutative 58.3%

         \[\leadsto 1 + \color{blue}{\frac{z}{y} \cdot -4} \]

      2. associate-*l/ 58.3%

         \[\leadsto 1 + \color{blue}{\frac{z \cdot -4}{y}} \]

   5. Simplified 58.3%

      \[\leadsto 1 + \color{blue}{\frac{z \cdot -4}{y}} \]

   if -9.4999999999999991e-56 < y < 8.49999999999999979e-285

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 66.3%

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

      1. associate-*r/ 66.3%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

   5. Simplified 66.3%

      \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.48 \cdot 10^{+53}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-56}:\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-285}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+79}:\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+84} \lor \neg \left(x \leq 5.3 \cdot 10^{+107}\right):\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e+84) (not (<= x 5.3e+107)))
   (+ 1.0 (/ (* 4.0 x) y))
   (+ 2.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e+84) || !(x <= 5.3e+107)) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d+84)) .or. (.not. (x <= 5.3d+107))) then
        tmp = 1.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 2.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e+84) || !(x <= 5.3e+107)) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.5e+84) or not (x <= 5.3e+107):
		tmp = 1.0 + ((4.0 * x) / y)
	else:
		tmp = 2.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5e+84) || !(x <= 5.3e+107))
		tmp = Float64(1.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(2.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.5e+84) || ~((x <= 5.3e+107)))
		tmp = 1.0 + ((4.0 * x) / y);
	else
		tmp = 2.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.5e+84], N[Not[LessEqual[x, 5.3e+107]], $MachinePrecision]], N[(1.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999998e84 or 5.3e107 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.5%

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

      1. associate-*r/ 79.5%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

   5. Simplified 79.5%

      \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

   if -1.49999999999999998e84 < x < 5.3e107

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{2 + \frac{4}{y} \cdot \left(x - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto 2 + \color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} \]

      2. associate-/l* 99.8%

         \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   7. Taylor expanded in x around 0 86.7%

      \[\leadsto 2 + \color{blue}{-4 \cdot \frac{z}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+84} \lor \neg \left(x \leq 5.3 \cdot 10^{+107}\right):\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.1 \cdot 10^{+62} \lor \neg \left(x \leq 1.05 \cdot 10^{-40}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.1e+62) (not (<= x 1.05e-40)))
   (+ 2.0 (* 4.0 (/ x y)))
   (+ 2.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.1e+62) || !(x <= 1.05e-40)) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.1d+62)) .or. (.not. (x <= 1.05d-40))) then
        tmp = 2.0d0 + (4.0d0 * (x / y))
    else
        tmp = 2.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.1e+62) || !(x <= 1.05e-40)) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.1e+62) or not (x <= 1.05e-40):
		tmp = 2.0 + (4.0 * (x / y))
	else:
		tmp = 2.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.1e+62) || !(x <= 1.05e-40))
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(2.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.1e+62) || ~((x <= 1.05e-40)))
		tmp = 2.0 + (4.0 * (x / y));
	else
		tmp = 2.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.1e+62], N[Not[LessEqual[x, 1.05e-40]], $MachinePrecision]], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.09999999999999998e62 or 1.05000000000000009e-40 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{2 + \frac{4}{y} \cdot \left(x - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto 2 + \color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} \]

      2. associate-/l* 99.8%

         \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   7. Taylor expanded in x around inf 87.5%

      \[\leadsto 2 + \frac{4}{\color{blue}{\frac{y}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 87.5%

         \[\leadsto 2 + \color{blue}{\frac{1}{\frac{\frac{y}{x}}{4}}} \]

      2. associate-/r/ 87.5%

         \[\leadsto 2 + \color{blue}{\frac{1}{\frac{y}{x}} \cdot 4} \]

      3. clear-num 87.6%

         \[\leadsto 2 + \color{blue}{\frac{x}{y}} \cdot 4 \]

   9. Applied egg-rr 87.6%

      \[\leadsto 2 + \color{blue}{\frac{x}{y} \cdot 4} \]

   if -8.09999999999999998e62 < x < 1.05000000000000009e-40

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{2 + \frac{4}{y} \cdot \left(x - z\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto 2 + \color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} \]

      2. associate-/l* 99.7%

         \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto 2 + \color{blue}{\frac{4}{\frac{y}{x - z}}} \]

   7. Taylor expanded in x around 0 92.3%

      \[\leadsto 2 + \color{blue}{-4 \cdot \frac{z}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.1 \cdot 10^{+62} \lor \neg \left(x \leq 1.05 \cdot 10^{-40}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-17}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e-17) 2.0 (if (<= y 8e-41) (+ 1.0 (* x (/ 4.0 y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e-17) {
		tmp = 2.0;
	} else if (y <= 8e-41) {
		tmp = 1.0 + (x * (4.0 / y));
	} else {
		tmp = 2.0;
	}
	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) :: tmp
    if (y <= (-4.7d-17)) then
        tmp = 2.0d0
    else if (y <= 8d-41) then
        tmp = 1.0d0 + (x * (4.0d0 / y))
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e-17) {
		tmp = 2.0;
	} else if (y <= 8e-41) {
		tmp = 1.0 + (x * (4.0 / y));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.7e-17:
		tmp = 2.0
	elif y <= 8e-41:
		tmp = 1.0 + (x * (4.0 / y))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e-17)
		tmp = 2.0;
	elseif (y <= 8e-41)
		tmp = Float64(1.0 + Float64(x * Float64(4.0 / y)));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.7e-17)
		tmp = 2.0;
	elseif (y <= 8e-41)
		tmp = 1.0 + (x * (4.0 / y));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.7e-17], 2.0, If[LessEqual[y, 8e-41], N[(1.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e-17 or 8.00000000000000005e-41 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.6%

      \[\leadsto \color{blue}{2} \]

   if -4.7e-17 < y < 8.00000000000000005e-41

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 52.0%

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

      1. associate-*r/ 52.0%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

      2. associate-*l/ 51.9%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot x} \]

      3. *-commutative 51.9%

         \[\leadsto 1 + \color{blue}{x \cdot \frac{4}{y}} \]

   5. Simplified 51.9%

      \[\leadsto 1 + \color{blue}{x \cdot \frac{4}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-17}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-14}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-40}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e-14) 2.0 (if (<= y 2.9e-40) (+ 1.0 (/ (* 4.0 x) y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-14) {
		tmp = 2.0;
	} else if (y <= 2.9e-40) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else {
		tmp = 2.0;
	}
	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) :: tmp
    if (y <= (-1.9d-14)) then
        tmp = 2.0d0
    else if (y <= 2.9d-40) then
        tmp = 1.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-14) {
		tmp = 2.0;
	} else if (y <= 2.9e-40) {
		tmp = 1.0 + ((4.0 * x) / y);
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.9e-14:
		tmp = 2.0
	elif y <= 2.9e-40:
		tmp = 1.0 + ((4.0 * x) / y)
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e-14)
		tmp = 2.0;
	elseif (y <= 2.9e-40)
		tmp = Float64(1.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.9e-14)
		tmp = 2.0;
	elseif (y <= 2.9e-40)
		tmp = 1.0 + ((4.0 * x) / y);
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e-14], 2.0, If[LessEqual[y, 2.9e-40], N[(1.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9000000000000001e-14 or 2.8999999999999999e-40 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.6%

      \[\leadsto \color{blue}{2} \]

   if -1.9000000000000001e-14 < y < 2.8999999999999999e-40

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 52.0%

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

      1. associate-*r/ 52.0%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

   5. Simplified 52.0%

      \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-14}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-40}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 2 + \frac{4}{y} \cdot \left(x - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 2.0 (* (/ 4.0 y) (- x z))))

C

double code(double x, double y, double z) {
	return 2.0 + ((4.0 / 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 = 2.0d0 + ((4.0d0 / y) * (x - z))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 + ((4.0 / y) * (x - z));
}

Python

def code(x, y, z):
	return 2.0 + ((4.0 / y) * (x - z))

Julia

function code(x, y, z)
	return Float64(2.0 + Float64(Float64(4.0 / y) * Float64(x - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 + ((4.0 / y) * (x - z));
end

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(N[(4.0 / y), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.8%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

   2. +-commutative 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

   3. associate--l+ 99.8%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

   4. distribute-lft-in 99.8%

      \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

   5. associate-+r+ 99.8%

      \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{2 + \frac{4}{y} \cdot \left(x - z\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto 2 + \frac{4}{y} \cdot \left(x - z\right) \]
6. Add Preprocessing

Alternative 8: 33.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 2.0)

C

double code(double x, double y, double z) {
	return 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 2.0;
}

Python

def code(x, y, z):
	return 2.0

Julia

function code(x, y, z)
	return 2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0;
end

Wolfram

code[x_, y_, z_] := 2.0

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 34.1%

   \[\leadsto \color{blue}{2} \]

4. Final simplification 34.1%

   \[\leadsto 2 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))