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

Percentage Accurate: 99.8% → 100.0%

Time: 10.5s

Alternatives: 9

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

5. Applied rewrites 100.0%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;\left(z - x\right) \cdot \frac{-4}{y}\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \left(x - z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_0 -10000.0)
     (* (- z x) (/ -4.0 y))
     (if (<= t_0 5.0) (fma (/ 4.0 y) x 4.0) (/ (* 4.0 (- x z)) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = (z - x) * (-4.0 / y);
	} else if (t_0 <= 5.0) {
		tmp = fma((4.0 / y), x, 4.0);
	} else {
		tmp = (4.0 * (x - z)) / y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(Float64(z - x) * Float64(-4.0 / y));
	elseif (t_0 <= 5.0)
		tmp = fma(Float64(4.0 / y), x, 4.0);
	else
		tmp = Float64(Float64(4.0 * Float64(x - z)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(N[(z - x), $MachinePrecision] * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision], N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e4

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 + \frac{4 \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)}{y}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)}{y} + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)}{y}} + 1 \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{4 \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)}}} + 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{4 \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)}}} + 1 \]

      6. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y}{4}}{\left(x + y \cdot \frac{3}{4}\right) - z}}} + 1 \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{4}} \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right)} + 1 \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{4}{y}} \cdot \left(\left(x + y \cdot \frac{3}{4}\right) - z\right) + 1 \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{y}, \left(x + y \cdot \frac{3}{4}\right) - z, 1\right)} \]

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{y}, \color{blue}{\left(x + y \cdot \frac{3}{4}\right)} - z, 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{y}, \color{blue}{\left(y \cdot \frac{3}{4} + x\right)} - z, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{4}{y}, \left(\color{blue}{y \cdot \frac{3}{4}} + x\right) - z, 1\right) \]

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 98.6%

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

   if -1e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   if 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} \]
3. Recombined 3 regimes into one program.
4. 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, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))