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

Percentage Accurate: 99.8% → 100.0%

Time: 6.1s

Alternatives: 6

Speedup: 1.2×

• 20.9% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(4.0 * N[(0.75 + N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $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* 99.9%

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

   2. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 97.7%

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

   1. associate--l+ 97.7%

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

   2. div-sub 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

   \[\leadsto 1 + 4 \cdot \left(0.75 + \frac{x - z}{y}\right) \]
9. Add Preprocessing

Alternative 2: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+79} \lor \neg \left(x \leq 3.9 \cdot 10^{+33}\right) \land \left(x \leq 6.2 \cdot 10^{+108} \lor \neg \left(x \leq 2.35 \cdot 10^{+132}\right)\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e+79)
         (and (not (<= x 3.9e+33))
              (or (<= x 6.2e+108) (not (<= x 2.35e+132)))))
   (+ 1.0 (* 4.0 (/ x y)))
   (+ 4.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+79) || (!(x <= 3.9e+33) && ((x <= 6.2e+108) || !(x <= 2.35e+132)))) {
		tmp = 1.0 + (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 <= (-2d+79)) .or. (.not. (x <= 3.9d+33)) .and. (x <= 6.2d+108) .or. (.not. (x <= 2.35d+132))) then
        tmp = 1.0d0 + (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 <= -2e+79) || (!(x <= 3.9e+33) && ((x <= 6.2e+108) || !(x <= 2.35e+132)))) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e+79) or (not (x <= 3.9e+33) and ((x <= 6.2e+108) or not (x <= 2.35e+132))):
		tmp = 1.0 + (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 <= -2e+79) || (!(x <= 3.9e+33) && ((x <= 6.2e+108) || !(x <= 2.35e+132))))
		tmp = Float64(1.0 + 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 <= -2e+79) || (~((x <= 3.9e+33)) && ((x <= 6.2e+108) || ~((x <= 2.35e+132)))))
		tmp = 1.0 + (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, -2e+79], And[N[Not[LessEqual[x, 3.9e+33]], $MachinePrecision], Or[LessEqual[x, 6.2e+108], N[Not[LessEqual[x, 2.35e+132]], $MachinePrecision]]]], N[(1.0 + 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 < -1.99999999999999993e79 or 3.9000000000000002e33 < x < 6.2000000000000003e108 or 2.35e132 < x

   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* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 86.3%

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

   if -1.99999999999999993e79 < x < 3.9000000000000002e33 or 6.2000000000000003e108 < x < 2.35e132

   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* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 90.1%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{0.75 \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub 90.1%

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

      2. associate-/l* 90.2%

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

      3. *-inverses 90.2%

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

      4. metadata-eval 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in z around 0 90.2%

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

      1. +-commutative 90.2%

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

   10. Simplified 90.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+79} \lor \neg \left(x \leq 3.9 \cdot 10^{+33}\right) \land \left(x \leq 6.2 \cdot 10^{+108} \lor \neg \left(x \leq 2.35 \cdot 10^{+132}\right)\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+78} \lor \neg \left(x \leq 1850000000000\right):\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e+78) (not (<= x 1850000000000.0)))
   (+ 1.0 (* 4.0 (/ (- x z) y)))
   (+ 4.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.5e+78) || !(x <= 1850000000000.0)) {
		tmp = 1.0 + (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 <= (-6.5d+78)) .or. (.not. (x <= 1850000000000.0d0))) then
        tmp = 1.0d0 + (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 <= -6.5e+78) || !(x <= 1850000000000.0)) {
		tmp = 1.0 + (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 <= -6.5e+78) or not (x <= 1850000000000.0):
		tmp = 1.0 + (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 <= -6.5e+78) || !(x <= 1850000000000.0))
		tmp = Float64(1.0 + 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 <= -6.5e+78) || ~((x <= 1850000000000.0)))
		tmp = 1.0 + (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, -6.5e+78], N[Not[LessEqual[x, 1850000000000.0]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(N[(x - z), $MachinePrecision] / 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 < -6.50000000000000036e78 or 1.85e12 < x

   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* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 92.2%

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

   if -6.50000000000000036e78 < x < 1.85e12

   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* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 90.2%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{0.75 \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub 90.3%

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

      2. associate-/l* 90.3%

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

      3. *-inverses 90.3%

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

      4. metadata-eval 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in z around 0 90.3%

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

      1. +-commutative 90.3%

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

   10. Simplified 90.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+78} \lor \neg \left(x \leq 1850000000000\right):\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+62} \lor \neg \left(x \leq 50000000000000\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.6e+62) (not (<= x 50000000000000.0)))
   (+ 1.0 (* 4.0 (/ x y)))
   4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.6e+62) || !(x <= 50000000000000.0)) {
		tmp = 1.0 + (4.0 * (x / 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 ((x <= (-7.6d+62)) .or. (.not. (x <= 50000000000000.0d0))) then
        tmp = 1.0d0 + (4.0d0 * (x / 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 ((x <= -7.6e+62) || !(x <= 50000000000000.0)) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.6e+62) or not (x <= 50000000000000.0):
		tmp = 1.0 + (4.0 * (x / y))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.6e+62) || !(x <= 50000000000000.0))
		tmp = Float64(1.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.6e+62) || ~((x <= 50000000000000.0)))
		tmp = 1.0 + (4.0 * (x / y));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.6e+62], N[Not[LessEqual[x, 50000000000000.0]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999967e62 or 5e13 < x

   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* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 78.2%

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

   if -7.59999999999999967e62 < x < 5e13

   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* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 90.1%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{0.75 \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub 90.1%

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

      2. associate-/l* 90.2%

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

      3. *-inverses 90.2%

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

      4. metadata-eval 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in z around 0 41.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+62} \lor \neg \left(x \leq 50000000000000\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.9%

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

   1. associate-/l* 99.9%

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

   2. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 42.4%

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

   1. neg-mul-1 42.4%

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

   2. distribute-neg-frac 42.4%

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

7. Simplified 42.4%

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

8. Taylor expanded in z around 0 6.7%

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

9. Final simplification 6.7%

   \[\leadsto 1 \]
10. Add Preprocessing

Alternative 6: 34.2% 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* 99.9%

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

   2. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 65.0%

   \[\leadsto 1 + 4 \cdot \color{blue}{\frac{0.75 \cdot y - z}{y}} \]
6. Step-by-step derivation

   1. div-sub 65.0%

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

   2. associate-/l* 65.1%

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

   3. *-inverses 65.1%

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

   4. metadata-eval 65.1%

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

7. Simplified 65.1%

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

8. Taylor expanded in z around 0 28.3%

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

9. Final simplification 28.3%

   \[\leadsto 4 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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)))