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

Percentage Accurate: 99.9% → 99.9%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

• 20.6% 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. Final simplification 100.0%

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

Alternative 2: 52.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y} \cdot -4\\ t_1 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -12200000:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ z y) -4.0))) (t_1 (* 4.0 (/ x y))))
   (if (<= y -12200000.0)
     2.0
     (if (<= y -8.2e-281)
       t_1
       (if (<= y 9.5e-248)
         t_0
         (if (<= y 3.6e-144)
           t_1
           (if (<= y 1.05e-97) t_0 (if (<= y 1.2e-8) t_1 2.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z / y) * -4.0);
	double t_1 = 4.0 * (x / y);
	double tmp;
	if (y <= -12200000.0) {
		tmp = 2.0;
	} else if (y <= -8.2e-281) {
		tmp = t_1;
	} else if (y <= 9.5e-248) {
		tmp = t_0;
	} else if (y <= 3.6e-144) {
		tmp = t_1;
	} else if (y <= 1.05e-97) {
		tmp = t_0;
	} else if (y <= 1.2e-8) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((z / y) * (-4.0d0))
    t_1 = 4.0d0 * (x / y)
    if (y <= (-12200000.0d0)) then
        tmp = 2.0d0
    else if (y <= (-8.2d-281)) then
        tmp = t_1
    else if (y <= 9.5d-248) then
        tmp = t_0
    else if (y <= 3.6d-144) then
        tmp = t_1
    else if (y <= 1.05d-97) then
        tmp = t_0
    else if (y <= 1.2d-8) then
        tmp = t_1
    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 / y) * -4.0);
	double t_1 = 4.0 * (x / y);
	double tmp;
	if (y <= -12200000.0) {
		tmp = 2.0;
	} else if (y <= -8.2e-281) {
		tmp = t_1;
	} else if (y <= 9.5e-248) {
		tmp = t_0;
	} else if (y <= 3.6e-144) {
		tmp = t_1;
	} else if (y <= 1.05e-97) {
		tmp = t_0;
	} else if (y <= 1.2e-8) {
		tmp = t_1;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((z / y) * -4.0)
	t_1 = 4.0 * (x / y)
	tmp = 0
	if y <= -12200000.0:
		tmp = 2.0
	elif y <= -8.2e-281:
		tmp = t_1
	elif y <= 9.5e-248:
		tmp = t_0
	elif y <= 3.6e-144:
		tmp = t_1
	elif y <= 1.05e-97:
		tmp = t_0
	elif y <= 1.2e-8:
		tmp = t_1
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(z / y) * -4.0))
	t_1 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (y <= -12200000.0)
		tmp = 2.0;
	elseif (y <= -8.2e-281)
		tmp = t_1;
	elseif (y <= 9.5e-248)
		tmp = t_0;
	elseif (y <= 3.6e-144)
		tmp = t_1;
	elseif (y <= 1.05e-97)
		tmp = t_0;
	elseif (y <= 1.2e-8)
		tmp = t_1;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((z / y) * -4.0);
	t_1 = 4.0 * (x / y);
	tmp = 0.0;
	if (y <= -12200000.0)
		tmp = 2.0;
	elseif (y <= -8.2e-281)
		tmp = t_1;
	elseif (y <= 9.5e-248)
		tmp = t_0;
	elseif (y <= 3.6e-144)
		tmp = t_1;
	elseif (y <= 1.05e-97)
		tmp = t_0;
	elseif (y <= 1.2e-8)
		tmp = t_1;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -12200000.0], 2.0, If[LessEqual[y, -8.2e-281], t$95$1, If[LessEqual[y, 9.5e-248], t$95$0, If[LessEqual[y, 3.6e-144], t$95$1, If[LessEqual[y, 1.05e-97], t$95$0, If[LessEqual[y, 1.2e-8], t$95$1, 2.0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.22e7 or 1.19999999999999999e-8 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 65.0%

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

   if -1.22e7 < y < -8.1999999999999998e-281 or 9.49999999999999971e-248 < y < 3.6e-144 or 1.0500000000000001e-97 < y < 1.19999999999999999e-8

   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. Taylor expanded in z around 0 67.7%

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

      1. +-commutative 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in x around inf 63.2%

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

   if -8.1999999999999998e-281 < y < 9.49999999999999971e-248 or 3.6e-144 < y < 1.0500000000000001e-97

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 71.3%

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

      1. *-commutative 71.3%

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

   4. Simplified 71.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12200000:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-281}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-248}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-144}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-97}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+213} \lor \neg \left(z \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.7e+213) (not (<= z 2.7e+81)))
   (+ 1.0 (* (/ z y) -4.0))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+213) || !(z <= 2.7e+81)) {
		tmp = 1.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / 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 ((z <= (-3.7d+213)) .or. (.not. (z <= 2.7d+81))) then
        tmp = 1.0d0 + ((z / y) * (-4.0d0))
    else
        tmp = 2.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+213) || !(z <= 2.7e+81)) {
		tmp = 1.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.7e+213) or not (z <= 2.7e+81):
		tmp = 1.0 + ((z / y) * -4.0)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.7e+213) || !(z <= 2.7e+81))
		tmp = Float64(1.0 + Float64(Float64(z / y) * -4.0));
	else
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.7e+213) || ~((z <= 2.7e+81)))
		tmp = 1.0 + ((z / y) * -4.0);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.7e+213], N[Not[LessEqual[z, 2.7e+81]], $MachinePrecision]], N[(1.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.69999999999999993e213 or 2.6999999999999999e81 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

   4. Simplified 84.7%

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

   if -3.69999999999999993e213 < z < 2.6999999999999999e81

   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/ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. associate-+r+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 88.1%

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

      1. +-commutative 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+213} \lor \neg \left(z \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 82.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+213} \lor \neg \left(z \leq 1.2 \cdot 10^{+78}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.7e+213) (not (<= z 1.2e+78)))
   (+ 2.0 (* (/ z y) -4.0))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+213) || !(z <= 1.2e+78)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / 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 ((z <= (-3.7d+213)) .or. (.not. (z <= 1.2d+78))) then
        tmp = 2.0d0 + ((z / y) * (-4.0d0))
    else
        tmp = 2.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+213) || !(z <= 1.2e+78)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.7e+213) or not (z <= 1.2e+78):
		tmp = 2.0 + ((z / y) * -4.0)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.7e+213) || !(z <= 1.2e+78))
		tmp = Float64(2.0 + Float64(Float64(z / y) * -4.0));
	else
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.7e+213) || ~((z <= 1.2e+78)))
		tmp = 2.0 + ((z / y) * -4.0);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.7e+213], N[Not[LessEqual[z, 1.2e+78]], $MachinePrecision]], N[(2.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.69999999999999993e213 or 1.1999999999999999e78 < z

   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/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--l+ 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. associate-+r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{2 + -4 \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. +-commutative 98.5%

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

      2. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   if -3.69999999999999993e213 < z < 1.1999999999999999e78

   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/ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. associate-+r+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 88.1%

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

      1. +-commutative 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+213} \lor \neg \left(z \leq 1.2 \cdot 10^{+78}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 52.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000000:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -16000000.0) 2.0 (if (<= y 1.4e-8) (* 4.0 (/ x y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -16000000.0) {
		tmp = 2.0;
	} else if (y <= 1.4e-8) {
		tmp = 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 <= (-16000000.0d0)) then
        tmp = 2.0d0
    else if (y <= 1.4d-8) then
        tmp = 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 <= -16000000.0) {
		tmp = 2.0;
	} else if (y <= 1.4e-8) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -16000000.0:
		tmp = 2.0
	elif y <= 1.4e-8:
		tmp = 4.0 * (x / y)
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -16000000.0)
		tmp = 2.0;
	elseif (y <= 1.4e-8)
		tmp = Float64(4.0 * Float64(x / y));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -16000000.0)
		tmp = 2.0;
	elseif (y <= 1.4e-8)
		tmp = 4.0 * (x / y);
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -16000000.0], 2.0, If[LessEqual[y, 1.4e-8], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e7 or 1.4e-8 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 65.0%

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

   if -1.6e7 < y < 1.4e-8

   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/ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate--l+ 98.9%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+r+ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 61.1%

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

      1. +-commutative 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in x around inf 57.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000000:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 6: 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/ 99.4%

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

   2. +-commutative 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. associate-+r+ 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 8.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 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 = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in z around inf 35.9%

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

   1. *-commutative 35.9%

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

4. Simplified 35.9%

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

5. Taylor expanded in z around 0 8.8%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 8.8%

   \[\leadsto 1 \]

Alternative 8: 33.5% 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. Taylor expanded in y around inf 39.2%

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

3. Final simplification 39.2%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023290 
(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)))