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

Percentage Accurate: 99.8% → 100.0%

Time: 7.1s

Alternatives: 9

Speedup: 1.5×

• 21.1% 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 99.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{y}\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) y)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -2e+147)
     t_0
     (if (<= t_1 -200.0) (* -4.0 (/ z y)) (if (<= t_1 1e+14) 2.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / y;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = -4.0 * (z / y);
	} else if (t_1 <= 1e+14) {
		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 * x) / y
    t_1 = ((((0.25d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-2d+147)) then
        tmp = t_0
    else if (t_1 <= (-200.0d0)) then
        tmp = (-4.0d0) * (z / y)
    else if (t_1 <= 1d+14) 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 * x) / y;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = -4.0 * (z / y);
	} else if (t_1 <= 1e+14) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * x) / y
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -2e+147:
		tmp = t_0
	elif t_1 <= -200.0:
		tmp = -4.0 * (z / y)
	elif t_1 <= 1e+14:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / y)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = Float64(-4.0 * Float64(z / y));
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * x) / y;
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = -4.0 * (z / y);
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.25 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+147], t$95$0, If[LessEqual[t$95$1, -200.0], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+14], 2.0, t$95$0]]]]]

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) < -2e147 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.3%

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

   if -2e147 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -200

   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

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 68.1

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

   8. Applied rewrites 68.1%

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

   if -200 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e14

   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. Applied rewrites 92.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -200:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{y} \cdot x\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 y) x)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -2e+147)
     t_0
     (if (<= t_1 -200.0) (* -4.0 (/ z y)) (if (<= t_1 1e+14) 2.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = -4.0 * (z / y);
	} else if (t_1 <= 1e+14) {
		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 / y) * x
    t_1 = ((((0.25d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-2d+147)) then
        tmp = t_0
    else if (t_1 <= (-200.0d0)) then
        tmp = (-4.0d0) * (z / y)
    else if (t_1 <= 1d+14) 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 / y) * x;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = -4.0 * (z / y);
	} else if (t_1 <= 1e+14) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / y) * x
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -2e+147:
		tmp = t_0
	elif t_1 <= -200.0:
		tmp = -4.0 * (z / y)
	elif t_1 <= 1e+14:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / y) * x)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = Float64(-4.0 * Float64(z / y));
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / y) * x;
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = -4.0 * (z / y);
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.25 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+147], t$95$0, If[LessEqual[t$95$1, -200.0], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+14], 2.0, t$95$0]]]]]

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) < -2e147 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -2e147 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -200

   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

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 68.1

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

   8. Applied rewrites 68.1%

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

   if -200 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e14

   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. Applied rewrites 92.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -200:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{y} \cdot x\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 y) x)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -2e+147)
     t_0
     (if (<= t_1 -200.0) (* (/ -4.0 y) z) (if (<= t_1 1e+14) 2.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = (-4.0 / y) * z;
	} else if (t_1 <= 1e+14) {
		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 / y) * x
    t_1 = ((((0.25d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-2d+147)) then
        tmp = t_0
    else if (t_1 <= (-200.0d0)) then
        tmp = ((-4.0d0) / y) * z
    else if (t_1 <= 1d+14) 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 / y) * x;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = t_0;
	} else if (t_1 <= -200.0) {
		tmp = (-4.0 / y) * z;
	} else if (t_1 <= 1e+14) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / y) * x
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -2e+147:
		tmp = t_0
	elif t_1 <= -200.0:
		tmp = (-4.0 / y) * z
	elif t_1 <= 1e+14:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / y) * x)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = Float64(Float64(-4.0 / y) * z);
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / y) * x;
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -2e+147)
		tmp = t_0;
	elseif (t_1 <= -200.0)
		tmp = (-4.0 / y) * z;
	elseif (t_1 <= 1e+14)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.25 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+147], t$95$0, If[LessEqual[t$95$1, -200.0], N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+14], 2.0, t$95$0]]]]]

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) < -2e147 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -2e147 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -200

   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 z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if -200 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e14

   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. Applied rewrites 92.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -200:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+14}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x - z}{y}\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ (- x z) y))) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -200.0) t_0 (if (<= t_1 1e+14) (fma (/ z y) -4.0 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * ((x - z) / y);
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+14) {
		tmp = fma((z / y), -4.0, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(Float64(x - z) / y))
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_0;
	elseif (t_1 <= 1e+14)
		tmp = fma(Float64(z / y), -4.0, 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.25 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$0, If[LessEqual[t$95$1, 1e+14], N[(N[(z / y), $MachinePrecision] * -4.0 + 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) < -200 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -200 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e14

   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

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 99.3

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

   8. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -200:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) z)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -200.0) 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 / y) * z;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -200.0) {
		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) / y) * z
    t_1 = ((((0.25d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-200.0d0)) 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 / y) * z;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -200.0) {
		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 / y) * z
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -200.0:
		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(Float64(-4.0 / y) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -200.0)
		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 / y) * z;
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -200.0)
		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[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.25 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], 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) < -200 or 2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 52.7

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

   5. Applied rewrites 52.7%

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

   if -200 < (/.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. Applied rewrites 97.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -200:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{z}{y}, -4, 2\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ z y) -4.0 2.0)))
   (if (<= z -1.55e-44) t_0 (if (<= z 3.2e+37) (fma (/ x y) 4.0 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z / y), -4.0, 2.0);
	double tmp;
	if (z <= -1.55e-44) {
		tmp = t_0;
	} else if (z <= 3.2e+37) {
		tmp = fma((x / y), 4.0, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z / y), -4.0, 2.0)
	tmp = 0.0
	if (z <= -1.55e-44)
		tmp = t_0;
	elseif (z <= 3.2e+37)
		tmp = fma(Float64(x / y), 4.0, 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] * -4.0 + 2.0), $MachinePrecision]}, If[LessEqual[z, -1.55e-44], t$95$0, If[LessEqual[z, 3.2e+37], N[(N[(x / y), $MachinePrecision] * 4.0 + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999992e-44 or 3.20000000000000014e37 < z

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 86.5

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

   8. Applied rewrites 86.5%

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

   if -1.54999999999999992e-44 < z < 3.20000000000000014e37

   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

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.3%

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

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ z y))))
   (if (<= z -1.25e+136) t_0 (if (<= z 8e+209) (fma (/ x y) 4.0 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (z / y);
	double tmp;
	if (z <= -1.25e+136) {
		tmp = t_0;
	} else if (z <= 8e+209) {
		tmp = fma((x / y), 4.0, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(z / y))
	tmp = 0.0
	if (z <= -1.25e+136)
		tmp = t_0;
	elseif (z <= 8e+209)
		tmp = fma(Float64(x / y), 4.0, 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]), $MachinePrecision]}, If[LessEqual[z, -1.25e+136], t$95$0, If[LessEqual[z, 8e+209], N[(N[(x / y), $MachinePrecision] * 4.0 + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e136 or 8.0000000000000006e209 < z

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -1.25e136 < z < 8.0000000000000006e209

   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

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.6%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+136}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.2% 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 99.6%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.0%

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

Reproduce

herbie shell --seed 2024277 
(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)))