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

Percentage Accurate: 99.8% → 100.0%

Time: 7.8s

Alternatives: 7

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

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

5. Simplified 99.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-/r/ N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. /-lowering-/.f64 N/A

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

   10. --lowering--.f64 100.0

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

7. Applied egg-rr 100.0%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) (fma 0.25 y 0.0)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -1000000000000.0)
     t_0
     (if (<= t_1 500000.0) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) / fma(0.25, y, 0.0);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) / fma(0.25, y, 0.0))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = fma(-4.0, Float64(z / y), 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[(0.25 * y + 0.0), $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, -1000000000000.0], t$95$0, If[LessEqual[t$95$1, 500000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 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) < -1e12 or 5e5 < (/.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. 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. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-rgt-identity N/A

          \[\leadsto \frac{x - z}{\color{blue}{\frac{1}{4} \cdot y + 0}} \]

      11. accelerator-lowering-fma.f64 99.1

          \[\leadsto \frac{x - z}{\color{blue}{\mathsf{fma}\left(0.25, y, 0\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}} \]

   if -1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5e5

   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

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-inverses N/A

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

      10. distribute-neg-frac N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. neg-mul-1 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. +-commutative N/A

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

      19. associate-+l+ N/A

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

   5. Simplified 99.2%

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

Alternative 3: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ z y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -50000.0) t_0 (if (<= t_1 2.0) 2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (z / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = (-4.0d0) * (z / y)
    t_1 = (4.0d0 * ((x + (y * 0.25d0)) - z)) / y
    if (t_1 <= (-50000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2.0d0) then
        tmp = 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (z / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (z / y)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	tmp = 0
	if t_1 <= -50000.0:
		tmp = t_0
	elif t_1 <= 2.0:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(z / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (z / y);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z / y), $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, -50000.0], t$95$0, If[LessEqual[t$95$1, 2.0], 2.0, 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) < -5e4 or 2 < (/.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 z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 54.6

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

   5. Simplified 54.6%

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

   if -5e4 < (/.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 y around inf

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

   5. Simplified 97.6%

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

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 2.0)))
   (if (<= z -5.5e-6) t_0 (if (<= z 3e+29) (fma 4.0 (/ x y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-4.0, (z / y), 2.0);
	double tmp;
	if (z <= -5.5e-6) {
		tmp = t_0;
	} else if (z <= 3e+29) {
		tmp = fma(4.0, (x / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-4.0, Float64(z / y), 2.0)
	tmp = 0.0
	if (z <= -5.5e-6)
		tmp = t_0;
	elseif (z <= 3e+29)
		tmp = fma(4.0, Float64(x / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[z, -5.5e-6], t$95$0, If[LessEqual[z, 3e+29], N[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999999e-6 or 2.9999999999999999e29 < 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

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-inverses N/A

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

      10. distribute-neg-frac N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. neg-mul-1 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. +-commutative N/A

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

      19. associate-+l+ N/A

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

   5. Simplified 89.2%

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

   if -5.4999999999999999e-6 < z < 2.9999999999999999e29

   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-*l* N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-rgt-identity N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. *-inverses N/A

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

   5. Simplified 94.0%

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

Alternative 5: 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 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

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

5. Simplified 99.8%

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

Alternative 6: 67.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. *-inverses N/A

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

   10. distribute-neg-frac N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

   16. neg-mul-1 N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. +-commutative N/A

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

   19. associate-+l+ N/A

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

5. Simplified 71.3%

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

Alternative 7: 33.7% 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. Simplified 36.1%

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

Reproduce

herbie shell --seed 2024196 
(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)))