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

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 7

Speedup: 1.0×

• 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: 99.9% 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]

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. Add Preprocessing

Alternative 2: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ t_1 := \frac{x}{y \cdot 0.25}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000000000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)) (t_1 (/ x (* y 0.25))))
   (if (<= t_0 -1e+97)
     (* -4.0 (/ z y))
     (if (<= t_0 -50.0) t_1 (if (<= t_0 50000000000.0) 2.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_1 = x / (y * 0.25);
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = -4.0 * (z / y);
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 2.0;
	} else {
		tmp = t_1;
	}
	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 * ((x + (y * 0.25d0)) - z)) / y
    t_1 = x / (y * 0.25d0)
    if (t_0 <= (-1d+97)) then
        tmp = (-4.0d0) * (z / y)
    else if (t_0 <= (-50.0d0)) then
        tmp = t_1
    else if (t_0 <= 50000000000.0d0) then
        tmp = 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_1 = x / (y * 0.25);
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = -4.0 * (z / y);
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.25)) - z)) / y
	t_1 = x / (y * 0.25)
	tmp = 0
	if t_0 <= -1e+97:
		tmp = -4.0 * (z / y)
	elif t_0 <= -50.0:
		tmp = t_1
	elif t_0 <= 50000000000.0:
		tmp = 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	t_1 = Float64(x / Float64(y * 0.25))
	tmp = 0.0
	if (t_0 <= -1e+97)
		tmp = Float64(-4.0 * Float64(z / y));
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 2.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	t_1 = x / (y * 0.25);
	tmp = 0.0;
	if (t_0 <= -1e+97)
		tmp = -4.0 * (z / y);
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+97], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 50000000000.0], 2.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1.0000000000000001e97

   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. lower-*.f64 N/A

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

      2. lower-/.f64 62.4

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

   5. Simplified 62.4%

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

   if -1.0000000000000001e97 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -50 or 5e10 < (/.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 x around inf

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 56.9

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

   5. Simplified 56.9%

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

   if -50 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5e10

   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 inf

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

   5. Simplified 92.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -1 \cdot 10^{+97}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -50:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 50000000000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -5e+18)
     t_0
     (if (<= t_1 50000000000.0) (fma -4.0 (/ z y) 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 <= -5e+18) {
		tmp = t_0;
	} else if (t_1 <= 50000000000.0) {
		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) / 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 <= -5e+18)
		tmp = t_0;
	elseif (t_1 <= 50000000000.0)
		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[(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, -5e+18], t$95$0, If[LessEqual[t$95$1, 50000000000.0], 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) < -5e18 or 5e10 < (/.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. 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 99.9

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

   5. Simplified 99.9%

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

   if -5e18 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5e10

   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 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 97.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x - z}{y \cdot 0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -50.0) 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 <= -50.0) {
		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 <= (-50.0d0)) 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 <= -50.0) {
		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 <= -50.0:
		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 <= -50.0)
		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 <= -50.0)
		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, -50.0], 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) < -50 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. lower-*.f64 N/A

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

      2. lower-/.f64 53.4

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

   5. Simplified 53.4%

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

   if -50 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 2

   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 inf

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

   5. Simplified 93.0%

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

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 2.0)))
   (if (<= z -2.8e+47) t_0 (if (<= z 3.8e-13) (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 <= -2.8e+47) {
		tmp = t_0;
	} else if (z <= 3.8e-13) {
		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 <= -2.8e+47)
		tmp = t_0;
	elseif (z <= 3.8e-13)
		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, -2.8e+47], t$95$0, If[LessEqual[z, 3.8e-13], N[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999988e47 or 3.8e-13 < 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 86.1%

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

   if -2.79999999999999988e47 < z < 3.8e-13

   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 93.7%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25))))
   (if (<= x -4.5e+176) t_0 (if (<= x 7e+170) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double tmp;
	if (x <= -4.5e+176) {
		tmp = t_0;
	} else if (x <= 7e+170) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(y * 0.25))
	tmp = 0.0
	if (x <= -4.5e+176)
		tmp = t_0;
	elseif (x <= 7e+170)
		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[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+176], t$95$0, If[LessEqual[x, 7e+170], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000003e176 or 7.00000000000000011e170 < 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. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.50000000000000003e176 < x < 7.00000000000000011e170

   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 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 82.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.8% 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 34.6%

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

Reproduce

herbie shell --seed 2024207 
(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)))