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

Percentage Accurate: 99.9% → 100.0%

Time: 3.8s

Alternatives: 7

Speedup: 1.4×

• 20.5% 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: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 2 + \frac{x - z}{y \cdot 0.25} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 2.0 (/ (- x z) (* y 0.25))))

C

double code(double x, double y, double z) {
	return 2.0 + ((x - z) / (y * 0.25));
}

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) / (y * 0.25d0))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 + ((x - z) / (y * 0.25));
}

Python

def code(x, y, z):
	return 2.0 + ((x - z) / (y * 0.25))

Julia

function code(x, y, z)
	return Float64(2.0 + Float64(Float64(x - z) / Float64(y * 0.25)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 + ((x - z) / (y * 0.25));
end

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(N[(x - z), $MachinePrecision] / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--l+ 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l* 99.8%

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

   9. metadata-eval 99.8%

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

   10. metadata-eval 99.8%

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

   11. associate-/l* 99.8%

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

   12. distribute-rgt-neg-in 99.8%

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

   13. distribute-lft-neg-out 99.8%

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

   14. associate-/l* 99.8%

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

   15. *-commutative 99.8%

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

   16. neg-mul-1 99.8%

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

   17. distribute-lft-neg-out 99.8%

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

   18. *-inverses 99.8%

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

   19. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

      \[\leadsto 2 + \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) \]

   4. associate-*l/ 100.0%

      \[\leadsto 2 + \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} \]

   5. *-un-lft-identity 100.0%

      \[\leadsto 2 + \frac{\color{blue}{x - z}}{y \cdot 0.25} \]

5. Applied egg-rr 100.0%

   \[\leadsto 2 + \color{blue}{\frac{x - z}{y \cdot 0.25}} \]

6. Final simplification 100.0%

   \[\leadsto 2 + \frac{x - z}{y \cdot 0.25} \]

Alternative 2: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-132} \lor \neg \left(x \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.4e-132) (not (<= x 1.9e-14))) (+ (* 4.0 (/ x y)) 1.0) 2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.4e-132) || !(x <= 1.9e-14)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} 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 ((x <= (-5.4d-132)) .or. (.not. (x <= 1.9d-14))) then
        tmp = (4.0d0 * (x / y)) + 1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.4e-132) || !(x <= 1.9e-14)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.4e-132) or not (x <= 1.9e-14):
		tmp = (4.0 * (x / y)) + 1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.4e-132) || !(x <= 1.9e-14))
		tmp = Float64(Float64(4.0 * Float64(x / y)) + 1.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.4e-132) || ~((x <= 1.9e-14)))
		tmp = (4.0 * (x / y)) + 1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.4e-132], N[Not[LessEqual[x, 1.9e-14]], $MachinePrecision]], N[(N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.3999999999999998e-132 or 1.9000000000000001e-14 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf 65.0%

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

   if -5.3999999999999998e-132 < x < 1.9000000000000001e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 50.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-132} \lor \neg \left(x \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 3: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-40} \lor \neg \left(z \leq 1.78 \cdot 10^{+196}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.5e-40) (not (<= z 1.78e+196)))
   (+ 2.0 (* (/ z y) -4.0))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-40) || !(z <= 1.78e+196)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8.5d-40)) .or. (.not. (z <= 1.78d+196))) then
        tmp = 2.0d0 + ((z / y) * (-4.0d0))
    else
        tmp = 2.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-40) || !(z <= 1.78e+196)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.5e-40) or not (z <= 1.78e+196):
		tmp = 2.0 + ((z / y) * -4.0)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.5e-40) || !(z <= 1.78e+196))
		tmp = Float64(2.0 + Float64(Float64(z / y) * -4.0));
	else
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.5e-40) || ~((z <= 1.78e+196)))
		tmp = 2.0 + ((z / y) * -4.0);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.5e-40], N[Not[LessEqual[z, 1.78e+196]], $MachinePrecision]], N[(2.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4999999999999998e-40 or 1.7799999999999999e196 < z

   1. Initial program 98.9%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-*l/ 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-/l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-/l* 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. distribute-lft-neg-out 99.8%

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

      14. associate-/l* 99.8%

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

      15. *-commutative 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-lft-neg-out 99.8%

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

      18. *-inverses 99.8%

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

      19. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. *-commutative 85.6%

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

   6. Simplified 85.6%

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

   if -8.4999999999999998e-40 < z < 1.7799999999999999e196

   1. Initial program 100.0%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-*l/ 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-/l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. associate-/l* 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. distribute-lft-neg-out 99.8%

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

      14. associate-/l* 99.8%

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

      15. *-commutative 99.8%

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

      16. neg-mul-1 99.8%

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

      17. distribute-lft-neg-out 99.8%

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

      18. *-inverses 99.8%

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

      19. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 90.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-40} \lor \neg \left(z \leq 1.78 \cdot 10^{+196}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

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 99.6%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--l+ 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l* 99.8%

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

   9. metadata-eval 99.8%

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

   10. metadata-eval 99.8%

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

   11. associate-/l* 99.8%

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

   12. distribute-rgt-neg-in 99.8%

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

   13. distribute-lft-neg-out 99.8%

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

   14. associate-/l* 99.8%

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

   15. *-commutative 99.8%

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

   16. neg-mul-1 99.8%

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

   17. distribute-lft-neg-out 99.8%

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

   18. *-inverses 99.8%

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

   19. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 67.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--l+ 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l* 99.8%

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

   9. metadata-eval 99.8%

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

   10. metadata-eval 99.8%

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

   11. associate-/l* 99.8%

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

   12. distribute-rgt-neg-in 99.8%

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

   13. distribute-lft-neg-out 99.8%

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

   14. associate-/l* 99.8%

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

   15. *-commutative 99.8%

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

   16. neg-mul-1 99.8%

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

   17. distribute-lft-neg-out 99.8%

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

   18. *-inverses 99.8%

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

   19. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around 0 71.7%

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

5. Final simplification 71.7%

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

Alternative 6: 8.2% 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%

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

2. Taylor expanded in x around inf 46.1%

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

3. Taylor expanded in x around 0 7.7%

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

4. Final simplification 7.7%

   \[\leadsto 1 \]

Alternative 7: 33.9% 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 99.6%

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

2. Taylor expanded in y around inf 32.5%

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

3. Final simplification 32.5%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023310 
(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)))