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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 8

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: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000000000:\\ \;\;\;\;2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25))) (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -100000000.0)
     t_0
     (if (<= t_1 10000000000.0)
       2.0
       (if (<= t_1 5e+97) (* -4.0 (/ z y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+97) {
		tmp = -4.0 * (z / y);
	} 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 = x / (y * 0.25d0)
    t_1 = (4.0d0 * ((x + (y * 0.25d0)) - z)) / y
    if (t_1 <= (-100000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 10000000000.0d0) then
        tmp = 2.0d0
    else if (t_1 <= 5d+97) then
        tmp = (-4.0d0) * (z / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+97) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (y * 0.25)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	tmp = 0
	if t_1 <= -100000000.0:
		tmp = t_0
	elif t_1 <= 10000000000.0:
		tmp = 2.0
	elif t_1 <= 5e+97:
		tmp = -4.0 * (z / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / 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 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+97)
		tmp = Float64(-4.0 * Float64(z / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (y * 0.25);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+97)
		tmp = -4.0 * (z / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / 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, -100000000.0], t$95$0, If[LessEqual[t$95$1, 10000000000.0], 2.0, If[LessEqual[t$95$1, 5e+97], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], 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) < -1e8 or 4.99999999999999999e97 < (/.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 99.6

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 58.3%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e10

   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 95.6%

      \[\leadsto \color{blue}{2} \]

   if 1e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 4.99999999999999999e97

   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 70.9

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

   5. Simplified 70.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{z}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000000000:\\ \;\;\;\;2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -100000000.0)
     t_0
     (if (<= t_1 10000000000.0)
       2.0
       (if (<= t_1 5e+97) (* -4.0 (/ z y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+97) {
		tmp = -4.0 * (z / y);
	} 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 = x * (4.0d0 / y)
    t_1 = (4.0d0 * ((x + (y * 0.25d0)) - z)) / y
    if (t_1 <= (-100000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 10000000000.0d0) then
        tmp = 2.0d0
    else if (t_1 <= 5d+97) then
        tmp = (-4.0d0) * (z / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+97) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	tmp = 0
	if t_1 <= -100000000.0:
		tmp = t_0
	elif t_1 <= 10000000000.0:
		tmp = 2.0
	elif t_1 <= 5e+97:
		tmp = -4.0 * (z / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+97)
		tmp = Float64(-4.0 * Float64(z / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+97)
		tmp = -4.0 * (z / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / 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, -100000000.0], t$95$0, If[LessEqual[t$95$1, 10000000000.0], 2.0, If[LessEqual[t$95$1, 5e+97], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], 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) < -1e8 or 4.99999999999999999e97 < (/.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 99.6

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 58.3%

       \[\leadsto \frac{\color{blue}{x}}{y \cdot 0.25} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

       7. metadata-eval 58.2

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

   12. Applied egg-rr 58.2%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e10

   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 95.6%

      \[\leadsto \color{blue}{2} \]

   if 1e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 4.99999999999999999e97

   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 70.9

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

   5. Simplified 70.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{z}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -100000000:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 10000000000:\\ \;\;\;\;2\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000000000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -1000000000000.0)
     t_0
     (if (<= t_1 10000000000.0) (fma 4.0 (/ x 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 <= -1000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = fma(4.0, (x / 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 <= -1000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		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[(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, -1000000000000.0], t$95$0, If[LessEqual[t$95$1, 10000000000.0], N[(4.0 * N[(x / 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 1e10 < (/.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 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if -1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e10

   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 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 99.0%

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

Alternative 5: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000000000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -100000000.0) t_0 (if (<= t_1 10000000000.0) 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 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		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 <= (-100000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 10000000000.0d0) 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 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		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 <= -100000000.0:
		tmp = t_0
	elif t_1 <= 10000000000.0:
		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 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		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 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		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, -100000000.0], t$95$0, If[LessEqual[t$95$1, 10000000000.0], 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) < -1e8 or 1e10 < (/.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 47.7

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

   5. Simplified 47.7%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 1e10

   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 95.6%

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

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 4.0 (/ x y) 2.0)))
   (if (<= x -2.2) t_0 (if (<= x 5.2e-67) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(4.0, (x / y), 2.0);
	double tmp;
	if (x <= -2.2) {
		tmp = t_0;
	} else if (x <= 5.2e-67) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(4.0, Float64(x / y), 2.0)
	tmp = 0.0
	if (x <= -2.2)
		tmp = t_0;
	elseif (x <= 5.2e-67)
		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[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[x, -2.2], t$95$0, If[LessEqual[x, 5.2e-67], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002 or 5.1999999999999998e-67 < 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 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 87.4%

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

   if -2.2000000000000002 < x < 5.1999999999999998e-67

   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 93.8%

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

Alternative 7: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ \mathbf{if}\;x \leq -850000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25))))
   (if (<= x -850000000000.0)
     t_0
     (if (<= x 3.7e+160) (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 <= -850000000000.0) {
		tmp = t_0;
	} else if (x <= 3.7e+160) {
		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 <= -850000000000.0)
		tmp = t_0;
	elseif (x <= 3.7e+160)
		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, -850000000000.0], t$95$0, If[LessEqual[x, 3.7e+160], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5e11 or 3.70000000000000016e160 < 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 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 86.4

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

   5. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{x - z}{\mathsf{fma}\left(0.25, y, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 86.4

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

   7. Applied egg-rr 86.4%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 79.2%

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

   if -8.5e11 < x < 3.70000000000000016e160

   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 85.2%

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

Alternative 8: 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 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 33.1%

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

Reproduce

herbie shell --seed 2024195 
(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)))