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

Percentage Accurate: 99.9% → 99.8%

Time: 9.7s

Alternatives: 7

Speedup: N/A×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.8%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -10000.0) t_0 (if (<= t_1 2.0) (fma (/ 4.0 y) x 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = fma((4.0 / y), x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) / Float64(y * 0.25))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(4.0 / y), x, 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$0, If[LessEqual[t$95$1, 2.0], N[(N[(4.0 / y), $MachinePrecision] * x + 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

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

   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 y around 0

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

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

   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-rgt-identity N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

      17. *-inverses N/A

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

      18. associate-*l* N/A

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

      19. associate-*l/ N/A

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

      20. *-lft-identity N/A

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

4. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2024228 
(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)))