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

Percentage Accurate: 99.7% → 100.0%

Time: 8.9s

Alternatives: 9

Speedup: 1.5×

• 21.3% 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: 99.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
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 99.8%

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

7. Applied rewrites 100.0%

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

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - z\right) \cdot 4}{y}\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- x z) 4.0) y)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -100000000.0)
     t_0
     (if (<= t_1 1e+14) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - z) * 4.0) / y;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+14) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - z) * 4.0) / y)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+14)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -100000000.0], t$95$0, If[LessEqual[t$95$1, 1e+14], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $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 3/4 binary64))) z)) y) < -1e8 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.7%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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 99.4%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e14

   1. Initial program 99.8%

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

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

   7. Applied rewrites 96.2%

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

   9. Applied rewrites 96.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -100000000:\\ \;\;\;\;\frac{\left(x - z\right) \cdot 4}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot 4}{y}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))