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

Percentage Accurate: 99.8% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.4×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - 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 * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - 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 * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 4 + 4 \cdot \frac{x - z}{y} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   3. fma-def 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

4. Taylor expanded in y around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto 4 + 4 \cdot \frac{x - z}{y} \]

Alternative 2: 54.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-246}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (* -4.0 (/ z y))))
   (if (<= x -2.65e+31)
     t_0
     (if (<= x -3.3e-43)
       t_1
       (if (<= x -8e-246)
         4.0
         (if (<= x 1.1e-302)
           t_1
           (if (<= x 1.5e-146) 4.0 (if (<= x 4.9e+68) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -2.65e+31) {
		tmp = t_0;
	} else if (x <= -3.3e-43) {
		tmp = t_1;
	} else if (x <= -8e-246) {
		tmp = 4.0;
	} else if (x <= 1.1e-302) {
		tmp = t_1;
	} else if (x <= 1.5e-146) {
		tmp = 4.0;
	} else if (x <= 4.9e+68) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    t_1 = (-4.0d0) * (z / y)
    if (x <= (-2.65d+31)) then
        tmp = t_0
    else if (x <= (-3.3d-43)) then
        tmp = t_1
    else if (x <= (-8d-246)) then
        tmp = 4.0d0
    else if (x <= 1.1d-302) then
        tmp = t_1
    else if (x <= 1.5d-146) then
        tmp = 4.0d0
    else if (x <= 4.9d+68) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -2.65e+31) {
		tmp = t_0;
	} else if (x <= -3.3e-43) {
		tmp = t_1;
	} else if (x <= -8e-246) {
		tmp = 4.0;
	} else if (x <= 1.1e-302) {
		tmp = t_1;
	} else if (x <= 1.5e-146) {
		tmp = 4.0;
	} else if (x <= 4.9e+68) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = -4.0 * (z / y)
	tmp = 0
	if x <= -2.65e+31:
		tmp = t_0
	elif x <= -3.3e-43:
		tmp = t_1
	elif x <= -8e-246:
		tmp = 4.0
	elif x <= 1.1e-302:
		tmp = t_1
	elif x <= 1.5e-146:
		tmp = 4.0
	elif x <= 4.9e+68:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(-4.0 * Float64(z / y))
	tmp = 0.0
	if (x <= -2.65e+31)
		tmp = t_0;
	elseif (x <= -3.3e-43)
		tmp = t_1;
	elseif (x <= -8e-246)
		tmp = 4.0;
	elseif (x <= 1.1e-302)
		tmp = t_1;
	elseif (x <= 1.5e-146)
		tmp = 4.0;
	elseif (x <= 4.9e+68)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = -4.0 * (z / y);
	tmp = 0.0;
	if (x <= -2.65e+31)
		tmp = t_0;
	elseif (x <= -3.3e-43)
		tmp = t_1;
	elseif (x <= -8e-246)
		tmp = 4.0;
	elseif (x <= 1.1e-302)
		tmp = t_1;
	elseif (x <= 1.5e-146)
		tmp = 4.0;
	elseif (x <= 4.9e+68)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.65e+31], t$95$0, If[LessEqual[x, -3.3e-43], t$95$1, If[LessEqual[x, -8e-246], 4.0, If[LessEqual[x, 1.1e-302], t$95$1, If[LessEqual[x, 1.5e-146], 4.0, If[LessEqual[x, 4.9e+68], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6500000000000002e31 or 4.89999999999999978e68 < x

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in x around inf 73.1%

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

   5. Taylor expanded in x around inf 70.5%

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

   if -2.6500000000000002e31 < x < -3.30000000000000016e-43 or -7.99999999999999965e-246 < x < 1.10000000000000004e-302 or 1.50000000000000009e-146 < x < 4.89999999999999978e68

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 91.8%

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

   6. Taylor expanded in z around inf 59.6%

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

   if -3.30000000000000016e-43 < x < -7.99999999999999965e-246 or 1.10000000000000004e-302 < x < 1.50000000000000009e-146

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 60.7%

      \[\leadsto \color{blue}{4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+31}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-246}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-302}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+68}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+30} \lor \neg \left(x \leq 1200000000000\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e+30) (not (<= x 1200000000000.0)))
   (* 4.0 (/ (- x z) y))
   (+ 4.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e+30) || !(x <= 1200000000000.0)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + (-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 <= (-2.9d+30)) .or. (.not. (x <= 1200000000000.0d0))) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0 + ((-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 <= -2.9e+30) || !(x <= 1200000000000.0)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e+30) or not (x <= 1200000000000.0):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e+30) || !(x <= 1200000000000.0))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e+30) || ~((x <= 1200000000000.0)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e+30], N[Not[LessEqual[x, 1200000000000.0]], $MachinePrecision]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e30 or 1.2e12 < x

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in y around 0 80.8%

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

   if -2.8999999999999998e30 < x < 1.2e12

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 94.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+30} \lor \neg \left(x \leq 1200000000000\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+27} \lor \neg \left(x \leq 34000000000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.1e+27) (not (<= x 34000000000000.0)))
   (+ 4.0 (/ (* 4.0 x) y))
   (+ 4.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e+27) || !(x <= 34000000000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + (-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 <= (-2.1d+27)) .or. (.not. (x <= 34000000000000.0d0))) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + ((-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 <= -2.1e+27) || !(x <= 34000000000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.1e+27) or not (x <= 34000000000000.0):
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.1e+27) || !(x <= 34000000000000.0))
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.1e+27) || ~((x <= 34000000000000.0)))
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.1e+27], N[Not[LessEqual[x, 34000000000000.0]], $MachinePrecision]], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999995e27 or 3.4e13 < x

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in z around 0 87.5%

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

      1. associate-*r/ 86.7%

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

   7. Simplified 86.7%

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

   if -2.09999999999999995e27 < x < 3.4e13

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 94.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+27} \lor \neg \left(x \leq 34000000000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+171}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+126}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+171) 4.0 (if (<= y 2.25e+126) (* 4.0 (/ (- x z) y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+171) {
		tmp = 4.0;
	} else if (y <= 2.25e+126) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	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 <= (-2.2d+171)) then
        tmp = 4.0d0
    else if (y <= 2.25d+126) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+171) {
		tmp = 4.0;
	} else if (y <= 2.25e+126) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e+171:
		tmp = 4.0
	elif y <= 2.25e+126:
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+171)
		tmp = 4.0;
	elseif (y <= 2.25e+126)
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e+171)
		tmp = 4.0;
	elseif (y <= 2.25e+126)
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+171], 4.0, If[LessEqual[y, 2.25e+126], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e171 or 2.24999999999999987e126 < y

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.5%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 83.4%

      \[\leadsto \color{blue}{4} \]

   if -2.1999999999999999e171 < y < 2.24999999999999987e126

   1. Initial program 99.5%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in y around 0 82.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+171}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+126}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 6: 51.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e+92) 4.0 (if (<= y 7.2e-77) (* -4.0 (/ z y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e+92) {
		tmp = 4.0;
	} else if (y <= 7.2e-77) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = 4.0;
	}
	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 <= (-1.5d+92)) then
        tmp = 4.0d0
    else if (y <= 7.2d-77) then
        tmp = (-4.0d0) * (z / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e+92) {
		tmp = 4.0;
	} else if (y <= 7.2e-77) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e+92:
		tmp = 4.0
	elif y <= 7.2e-77:
		tmp = -4.0 * (z / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e+92)
		tmp = 4.0;
	elseif (y <= 7.2e-77)
		tmp = Float64(-4.0 * Float64(z / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.5e+92)
		tmp = 4.0;
	elseif (y <= 7.2e-77)
		tmp = -4.0 * (z / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e+92], 4.0, If[LessEqual[y, 7.2e-77], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.50000000000000007e92 or 7.2e-77 < y

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around inf 59.6%

      \[\leadsto \color{blue}{4} \]

   if -1.50000000000000007e92 < y < 7.2e-77

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around 0 58.0%

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

   6. Taylor expanded in z around inf 48.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 7: 7.6% 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 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   3. fma-def 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

4. Taylor expanded in x around inf 41.9%

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

5. Taylor expanded in x around 0 7.7%

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

6. Final simplification 7.7%

   \[\leadsto 1 \]

Alternative 8: 33.5% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   3. fma-def 99.7%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]

4. Taylor expanded in y around inf 35.0%

   \[\leadsto \color{blue}{4} \]

5. Final simplification 35.0%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))