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

Percentage Accurate: 99.6% → 99.9%

Time: 8.3s

Alternatives: 6

Speedup: 1.4×

• 20.1% 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: 99.6% 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: 99.9% accurate, 1.4× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

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/ 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-define 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. Add Preprocessing

5. Taylor expanded in y around 0 100.0%

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

   1. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+174} \lor \neg \left(z \leq -1.85 \cdot 10^{+61}\right) \land \left(z \leq -0.000135 \lor \neg \left(z \leq 1.5 \cdot 10^{+91} \lor \neg \left(z \leq 3.7 \cdot 10^{+111}\right) \land z \leq 1.06 \cdot 10^{+171}\right)\right):\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.6e+174)
         (and (not (<= z -1.85e+61))
              (or (<= z -0.000135)
                  (not
                   (or (<= z 1.5e+91)
                       (and (not (<= z 3.7e+111)) (<= z 1.06e+171)))))))
   (+ (* -4.0 (/ z y)) 1.0)
   (+ 4.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.6e+174) || (!(z <= -1.85e+61) && ((z <= -0.000135) || !((z <= 1.5e+91) || (!(z <= 3.7e+111) && (z <= 1.06e+171)))))) {
		tmp = (-4.0 * (z / y)) + 1.0;
	} else {
		tmp = 4.0 + (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 <= (-3.6d+174)) .or. (.not. (z <= (-1.85d+61))) .and. (z <= (-0.000135d0)) .or. (.not. (z <= 1.5d+91) .or. (.not. (z <= 3.7d+111)) .and. (z <= 1.06d+171))) then
        tmp = ((-4.0d0) * (z / y)) + 1.0d0
    else
        tmp = 4.0d0 + (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 <= -3.6e+174) || (!(z <= -1.85e+61) && ((z <= -0.000135) || !((z <= 1.5e+91) || (!(z <= 3.7e+111) && (z <= 1.06e+171)))))) {
		tmp = (-4.0 * (z / y)) + 1.0;
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.6e+174) or (not (z <= -1.85e+61) and ((z <= -0.000135) or not ((z <= 1.5e+91) or (not (z <= 3.7e+111) and (z <= 1.06e+171))))):
		tmp = (-4.0 * (z / y)) + 1.0
	else:
		tmp = 4.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.6e+174) || (!(z <= -1.85e+61) && ((z <= -0.000135) || !((z <= 1.5e+91) || (!(z <= 3.7e+111) && (z <= 1.06e+171))))))
		tmp = Float64(Float64(-4.0 * Float64(z / y)) + 1.0);
	else
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.6e+174) || (~((z <= -1.85e+61)) && ((z <= -0.000135) || ~(((z <= 1.5e+91) || (~((z <= 3.7e+111)) && (z <= 1.06e+171)))))))
		tmp = (-4.0 * (z / y)) + 1.0;
	else
		tmp = 4.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.6e+174], And[N[Not[LessEqual[z, -1.85e+61]], $MachinePrecision], Or[LessEqual[z, -0.000135], N[Not[Or[LessEqual[z, 1.5e+91], And[N[Not[LessEqual[z, 3.7e+111]], $MachinePrecision], LessEqual[z, 1.06e+171]]]], $MachinePrecision]]]], N[(N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6000000000000002e174 or -1.85000000000000001e61 < z < -1.35000000000000002e-4 or 1.50000000000000003e91 < z < 3.7000000000000003e111 or 1.06000000000000001e171 < z

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in z around inf 81.5%

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

      1. *-commutative 81.5%

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

   7. Simplified 81.5%

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

   if -3.6000000000000002e174 < z < -1.85000000000000001e61 or -1.35000000000000002e-4 < z < 1.50000000000000003e91 or 3.7000000000000003e111 < z < 1.06000000000000001e171

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 85.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+174} \lor \neg \left(z \leq -1.85 \cdot 10^{+61}\right) \land \left(z \leq -0.000135 \lor \neg \left(z \leq 1.5 \cdot 10^{+91} \lor \neg \left(z \leq 3.7 \cdot 10^{+111}\right) \land z \leq 1.06 \cdot 10^{+171}\right)\right):\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y} + 1\\ t_1 := 4 \cdot \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -6000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-48}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* -4.0 (/ z y)) 1.0)) (t_1 (+ (* 4.0 (/ x y)) 1.0)))
   (if (<= x -6000000000000.0)
     t_1
     (if (<= x 1.35e-260)
       4.0
       (if (<= x 6.2e-92)
         t_0
         (if (<= x 3.3e-48) 4.0 (if (<= x 1.5e+82) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * (z / y)) + 1.0;
	double t_1 = (4.0 * (x / y)) + 1.0;
	double tmp;
	if (x <= -6000000000000.0) {
		tmp = t_1;
	} else if (x <= 1.35e-260) {
		tmp = 4.0;
	} else if (x <= 6.2e-92) {
		tmp = t_0;
	} else if (x <= 3.3e-48) {
		tmp = 4.0;
	} else if (x <= 1.5e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) * (z / y)) + 1.0d0
    t_1 = (4.0d0 * (x / y)) + 1.0d0
    if (x <= (-6000000000000.0d0)) then
        tmp = t_1
    else if (x <= 1.35d-260) then
        tmp = 4.0d0
    else if (x <= 6.2d-92) then
        tmp = t_0
    else if (x <= 3.3d-48) then
        tmp = 4.0d0
    else if (x <= 1.5d+82) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-4.0 * (z / y)) + 1.0;
	double t_1 = (4.0 * (x / y)) + 1.0;
	double tmp;
	if (x <= -6000000000000.0) {
		tmp = t_1;
	} else if (x <= 1.35e-260) {
		tmp = 4.0;
	} else if (x <= 6.2e-92) {
		tmp = t_0;
	} else if (x <= 3.3e-48) {
		tmp = 4.0;
	} else if (x <= 1.5e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 * (z / y)) + 1.0
	t_1 = (4.0 * (x / y)) + 1.0
	tmp = 0
	if x <= -6000000000000.0:
		tmp = t_1
	elif x <= 1.35e-260:
		tmp = 4.0
	elif x <= 6.2e-92:
		tmp = t_0
	elif x <= 3.3e-48:
		tmp = 4.0
	elif x <= 1.5e+82:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * Float64(z / y)) + 1.0)
	t_1 = Float64(Float64(4.0 * Float64(x / y)) + 1.0)
	tmp = 0.0
	if (x <= -6000000000000.0)
		tmp = t_1;
	elseif (x <= 1.35e-260)
		tmp = 4.0;
	elseif (x <= 6.2e-92)
		tmp = t_0;
	elseif (x <= 3.3e-48)
		tmp = 4.0;
	elseif (x <= 1.5e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 * (z / y)) + 1.0;
	t_1 = (4.0 * (x / y)) + 1.0;
	tmp = 0.0;
	if (x <= -6000000000000.0)
		tmp = t_1;
	elseif (x <= 1.35e-260)
		tmp = 4.0;
	elseif (x <= 6.2e-92)
		tmp = t_0;
	elseif (x <= 3.3e-48)
		tmp = 4.0;
	elseif (x <= 1.5e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -6000000000000.0], t$95$1, If[LessEqual[x, 1.35e-260], 4.0, If[LessEqual[x, 6.2e-92], t$95$0, If[LessEqual[x, 3.3e-48], 4.0, If[LessEqual[x, 1.5e+82], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e12 or 1.49999999999999995e82 < x

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.9%

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

   if -6e12 < x < 1.35000000000000003e-260 or 6.2000000000000002e-92 < x < 3.3e-48

   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-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 69.7%

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

   if 1.35000000000000003e-260 < x < 6.2000000000000002e-92 or 3.3e-48 < x < 1.49999999999999995e82

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in z around inf 68.8%

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

      1. *-commutative 68.8%

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

   7. Simplified 68.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6000000000000:\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-92}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-48}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+14} \lor \neg \left(x \leq 4.1 \cdot 10^{-44}\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.6e+14) (not (<= x 4.1e-44))) (+ (* 4.0 (/ x y)) 1.0) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e+14) || !(x <= 4.1e-44)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} 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 ((x <= (-2.6d+14)) .or. (.not. (x <= 4.1d-44))) then
        tmp = (4.0d0 * (x / y)) + 1.0d0
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e+14) || !(x <= 4.1e-44)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.6e+14) or not (x <= 4.1e-44):
		tmp = (4.0 * (x / y)) + 1.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.6e+14) || !(x <= 4.1e-44))
		tmp = Float64(Float64(4.0 * Float64(x / y)) + 1.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.6e+14) || ~((x <= 4.1e-44)))
		tmp = (4.0 * (x / y)) + 1.0;
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.6e+14], N[Not[LessEqual[x, 4.1e-44]], $MachinePrecision]], N[(N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e14 or 4.09999999999999992e-44 < x

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 60.3%

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

   if -2.6e14 < x < 4.09999999999999992e-44

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 56.8%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+14} \lor \neg \left(x \leq 4.1 \cdot 10^{-44}\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-55} \lor \neg \left(x \leq 3.1 \cdot 10^{+77}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4e-55) (not (<= x 3.1e+77)))
   (+ 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 <= -1.4e-55) || !(x <= 3.1e+77)) {
		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 <= (-1.4d-55)) .or. (.not. (x <= 3.1d+77))) 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 <= -1.4e-55) || !(x <= 3.1e+77)) {
		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 <= -1.4e-55) or not (x <= 3.1e+77):
		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 <= -1.4e-55) || !(x <= 3.1e+77))
		tmp = Float64(4.0 + Float64(4.0 * Float64(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 <= -1.4e-55) || ~((x <= 3.1e+77)))
		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, -1.4e-55], N[Not[LessEqual[x, 3.1e+77]], $MachinePrecision]], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999992e-55 or 3.09999999999999999e77 < x

   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/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 83.8%

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

   if -1.39999999999999992e-55 < x < 3.09999999999999999e77

   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-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 93.3%

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

      1. +-commutative 93.3%

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

   10. Simplified 93.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-55} \lor \neg \left(x \leq 3.1 \cdot 10^{+77}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.5% 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 99.9%

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

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. fma-define 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. Add Preprocessing

5. Taylor expanded in y around inf 38.7%

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

6. Final simplification 38.7%

   \[\leadsto 4 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))