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

Percentage Accurate: 99.8% → 100.0%

Time: 7.1s

Alternatives: 6

Speedup: 1.5×

• 20.2% 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: 99.8% 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, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fma(((x - z) / y), 4.0, 2.0);
}

Julia

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

Wolfram

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

3. Taylor expanded in x around 0

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

   1. *-inverses N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r/ N/A

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

   4. div-add-rev N/A

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

   5. distribute-lft-out N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. distribute-lft-out N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. div-add N/A

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

   14. associate-+l+ N/A

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

   15. count-2-rev N/A

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

   16. +-commutative N/A

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)} \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17} \lor \neg \left(t\_0 \leq 10000000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y))))
   (if (or (<= t_0 -1e+17) (not (<= t_0 10000000000000.0)))
     (* (/ (- x z) y) 4.0)
     (fma 4.0 (/ x y) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
	double tmp;
	if ((t_0 <= -1e+17) || !(t_0 <= 10000000000000.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma(4.0, (x / y), 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
	tmp = 0.0
	if ((t_0 <= -1e+17) || !(t_0 <= 10000000000000.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(4.0, Float64(x / y), 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+17], N[Not[LessEqual[t$95$0, 10000000000000.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)) < -1e17 or 1e13 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e17 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)) < 1e13

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{y}, x, 2\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{y}}, x, 2\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{y}, x, 2\right) \]

      16. lower-/.f64 97.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{y}}, x, 2\right) \]

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{y}}, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -1 \cdot 10^{+17} \lor \neg \left(1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 10000000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -2 \lor \neg \left(t\_0 \leq 10000\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y))))
   (if (or (<= t_0 -2.0) (not (<= t_0 10000.0))) (* (/ z y) -4.0) 2.0)))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 10000.0)) {
		tmp = (z / y) * -4.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((4.0d0 * ((x + (y * 0.25d0)) - z)) / y)
    if ((t_0 <= (-2.0d0)) .or. (.not. (t_0 <= 10000.0d0))) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 10000.0)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)
	tmp = 0
	if (t_0 <= -2.0) or not (t_0 <= 10000.0):
		tmp = (z / y) * -4.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
	tmp = 0.0
	if ((t_0 <= -2.0) || !(t_0 <= 10000.0))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
	tmp = 0.0;
	if ((t_0 <= -2.0) || ~((t_0 <= 10000.0)))
		tmp = (z / y) * -4.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2.0], N[Not[LessEqual[t$95$0, 10000.0]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)) < -2 or 1e4 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.3%

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

   if -2 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)) < 1e4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -2 \lor \neg \left(1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 10000\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-23} \lor \neg \left(z \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e-23) (not (<= z 2.2e-11)))
   (fma (/ z y) -4.0 2.0)
   (fma 4.0 (/ x y) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-23) || !(z <= 2.2e-11)) {
		tmp = fma((z / y), -4.0, 2.0);
	} else {
		tmp = fma(4.0, (x / y), 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e-23) || !(z <= 2.2e-11))
		tmp = fma(Float64(z / y), -4.0, 2.0);
	else
		tmp = fma(4.0, Float64(x / y), 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e-23], N[Not[LessEqual[z, 2.2e-11]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0 + 2.0), $MachinePrecision], N[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2500000000000001e-23 or 2.2000000000000002e-11 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. distribute-lft-out N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. distribute-lft-out N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. div-add N/A

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

      14. associate-+l+ N/A

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

      15. count-2-rev N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{y}, \color{blue}{-4}, 2\right) \]

   if -1.2500000000000001e-23 < z < 2.2000000000000002e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{y}, x, 2\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{y}}, x, 2\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{y}, x, 2\right) \]

      16. lower-/.f64 96.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{y}}, x, 2\right) \]

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{y}}, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-23} \lor \neg \left(z \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+157)
   (* (/ x y) 4.0)
   (if (<= x 1.85e+165) (fma (/ z y) -4.0 2.0) (* x (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+157) {
		tmp = (x / y) * 4.0;
	} else if (x <= 1.85e+165) {
		tmp = fma((z / y), -4.0, 2.0);
	} else {
		tmp = x * (4.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+157)
		tmp = Float64(Float64(x / y) * 4.0);
	elseif (x <= 1.85e+165)
		tmp = fma(Float64(z / y), -4.0, 2.0);
	else
		tmp = Float64(x * Float64(4.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e+157], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[x, 1.85e+165], N[(N[(z / y), $MachinePrecision] * -4.0 + 2.0), $MachinePrecision], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e157

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -1.2e157 < x < 1.85000000000000003e165

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. distribute-lft-out N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. distribute-lft-out N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. div-add N/A

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

      14. associate-+l+ N/A

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

      15. count-2-rev N/A

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

      16. +-commutative N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.2%

      \[\leadsto \mathsf{fma}\left(\frac{z}{y}, \color{blue}{-4}, 2\right) \]

   if 1.85000000000000003e165 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 84.4%

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

Alternative 6: 34.5% accurate, 31.0× 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. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{2} \]
4. Step-by-step derivation

5. Applied rewrites 36.7%

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

Reproduce

herbie shell --seed 2024337 
(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)))