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

Percentage Accurate: 99.9% → 99.9%

Time: 4.3s

Alternatives: 6

Speedup: 1.4×

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

Specification

Math

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

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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + 2.0), $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. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. associate-+l+ 99.8%

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

   8. associate-*l/ 100.0%

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

   9. associate-*l/ 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. *-inverses 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1250000000 \lor \neg \left(x \leq 1.25 \cdot 10^{-68}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1250000000.0) (not (<= x 1.25e-68)))
   (+ 2.0 (* 4.0 (/ x y)))
   (+ 2.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1250000000.0) || !(x <= 1.25e-68)) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1250000000.0d0)) .or. (.not. (x <= 1.25d-68))) then
        tmp = 2.0d0 + (4.0d0 * (x / y))
    else
        tmp = 2.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1250000000.0) || !(x <= 1.25e-68)) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1250000000.0) or not (x <= 1.25e-68):
		tmp = 2.0 + (4.0 * (x / y))
	else:
		tmp = 2.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1250000000.0) || !(x <= 1.25e-68))
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(2.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1250000000.0) || ~((x <= 1.25e-68)))
		tmp = 2.0 + (4.0 * (x / y));
	else
		tmp = 2.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1250000000.0], N[Not[LessEqual[x, 1.25e-68]], $MachinePrecision]], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e9 or 1.24999999999999993e-68 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l/ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. associate-+l+ 99.7%

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

      8. associate-*l/ 100.0%

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

      9. associate-*l/ 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-rgt-identity 100.0%

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

      14. *-inverses 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 84.0%

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

   if -1.25e9 < x < 1.24999999999999993e-68

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. associate-+l+ 99.9%

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

      8. associate-*l/ 100.0%

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

      9. associate-*l/ 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-rgt-identity 100.0%

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

      14. *-inverses 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 93.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1250000000 \lor \neg \left(x \leq 1.25 \cdot 10^{-68}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4300000000000:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+85}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4300000000000.0) 2.0 (if (<= y 4.1e+85) (* -4.0 (/ z y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4300000000000.0) {
		tmp = 2.0;
	} else if (y <= 4.1e+85) {
		tmp = -4.0 * (z / 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 <= (-4300000000000.0d0)) then
        tmp = 2.0d0
    else if (y <= 4.1d+85) then
        tmp = (-4.0d0) * (z / 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 <= -4300000000000.0) {
		tmp = 2.0;
	} else if (y <= 4.1e+85) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4300000000000.0:
		tmp = 2.0
	elif y <= 4.1e+85:
		tmp = -4.0 * (z / y)
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4300000000000.0)
		tmp = 2.0;
	elseif (y <= 4.1e+85)
		tmp = Float64(-4.0 * Float64(z / y));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4300000000000.0)
		tmp = 2.0;
	elseif (y <= 4.1e+85)
		tmp = -4.0 * (z / y);
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4300000000000.0], 2.0, If[LessEqual[y, 4.1e+85], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3e12 or 4.09999999999999978e85 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. associate-+l+ 99.8%

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

      8. associate-*l/ 100.0%

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

      9. associate-*l/ 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-rgt-identity 100.0%

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

      14. *-inverses 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 81.8%

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

   6. Taylor expanded in z around 0 65.4%

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

   if -4.3e12 < y < 4.09999999999999978e85

   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. +-commutative 100.0%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. associate-+l+ 99.8%

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

      8. associate-*l/ 100.0%

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

      9. associate-*l/ 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-rgt-identity 100.0%

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

      14. *-inverses 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 53.6%

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

   6. Taylor expanded in z around inf 47.5%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $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. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. associate-+l+ 99.8%

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

   8. associate-*l/ 100.0%

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

   9. associate-*l/ 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. *-inverses 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-*l/ 97.3%

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

   3. associate-*r/ 97.2%

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

   4. metadata-eval 97.2%

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

   5. distribute-neg-frac 97.2%

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

   6. metadata-eval 97.2%

      \[\leadsto \left(z \cdot \left(-\frac{\color{blue}{4 \cdot 1}}{y}\right) + 4 \cdot \frac{x}{y}\right) + 2 \]

   7. associate-*r/ 97.2%

      \[\leadsto \left(z \cdot \left(-\color{blue}{4 \cdot \frac{1}{y}}\right) + 4 \cdot \frac{x}{y}\right) + 2 \]

   8. distribute-rgt-neg-in 97.2%

      \[\leadsto \left(\color{blue}{\left(-z \cdot \left(4 \cdot \frac{1}{y}\right)\right)} + 4 \cdot \frac{x}{y}\right) + 2 \]

   9. *-commutative 97.2%

      \[\leadsto \left(\left(-\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right) + 4 \cdot \frac{x}{y}\right) + 2 \]

   10. distribute-rgt-neg-in 97.2%

       \[\leadsto \left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot \left(-z\right)} + 4 \cdot \frac{x}{y}\right) + 2 \]

   11. associate-*r/ 97.2%

       \[\leadsto \left(\left(4 \cdot \frac{1}{y}\right) \cdot \left(-z\right) + \color{blue}{\frac{4 \cdot x}{y}}\right) + 2 \]

   12. associate-*l/ 97.1%

       \[\leadsto \left(\left(4 \cdot \frac{1}{y}\right) \cdot \left(-z\right) + \color{blue}{\frac{4}{y} \cdot x}\right) + 2 \]

   13. metadata-eval 97.1%

       \[\leadsto \left(\left(4 \cdot \frac{1}{y}\right) \cdot \left(-z\right) + \frac{\color{blue}{4 \cdot 1}}{y} \cdot x\right) + 2 \]

   14. associate-*r/ 97.1%

       \[\leadsto \left(\left(4 \cdot \frac{1}{y}\right) \cdot \left(-z\right) + \color{blue}{\left(4 \cdot \frac{1}{y}\right)} \cdot x\right) + 2 \]

   15. distribute-lft-in 99.8%

       \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot \left(\left(-z\right) + x\right)} + 2 \]

   16. +-commutative 99.8%

       \[\leadsto \left(4 \cdot \frac{1}{y}\right) \cdot \color{blue}{\left(x + \left(-z\right)\right)} + 2 \]

   17. sub-neg 99.8%

       \[\leadsto \left(4 \cdot \frac{1}{y}\right) \cdot \color{blue}{\left(x - z\right)} + 2 \]

   18. associate-*r/ 99.8%

       \[\leadsto \color{blue}{\frac{4 \cdot 1}{y}} \cdot \left(x - z\right) + 2 \]

   19. metadata-eval 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 + -4 \cdot \frac{z}{y} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(-4.0 * N[(z / y), $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. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. associate-+l+ 99.8%

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

   8. associate-*l/ 100.0%

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

   9. associate-*l/ 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. *-inverses 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 64.7%

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

6. Final simplification 64.7%

   \[\leadsto 2 + -4 \cdot \frac{z}{y} \]
7. Add Preprocessing

Alternative 6: 35.2% 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. associate-+l+ 99.8%

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

   8. associate-*l/ 100.0%

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

   9. associate-*l/ 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-rgt-identity 100.0%

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

   14. *-inverses 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 64.7%

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

6. Taylor expanded in z around 0 31.1%

   \[\leadsto \color{blue}{2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(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)))