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

Percentage Accurate: 99.9% → 100.0%

Time: 7.8s

Alternatives: 10

Speedup: 1.5×

• 20.7% 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: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0 + 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 65.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -100000:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z y) -4.0)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -2e+53)
     t_0
     (if (<= t_1 -100000.0) (* (/ x y) 4.0) (if (<= t_1 30000.0) 2.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (x / y) * 4.0;
	} else if (t_1 <= 30000.0) {
		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 = (z / y) * (-4.0d0)
    t_1 = ((((0.25d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-2d+53)) then
        tmp = t_0
    else if (t_1 <= (-100000.0d0)) then
        tmp = (x / y) * 4.0d0
    else if (t_1 <= 30000.0d0) 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 = (z / y) * -4.0;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (x / y) * 4.0;
	} else if (t_1 <= 30000.0) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) * -4.0
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -2e+53:
		tmp = t_0
	elif t_1 <= -100000.0:
		tmp = (x / y) * 4.0
	elif t_1 <= 30000.0:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / y) * -4.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = Float64(Float64(x / y) * 4.0);
	elseif (t_1 <= 30000.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) * -4.0;
	t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = (x / y) * 4.0;
	elseif (t_1 <= 30000.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] * -4.0), $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+53], t$95$0, If[LessEqual[t$95$1, -100000.0], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[t$95$1, 30000.0], 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) < -2e53 or 3e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

   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 62.4

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

   8. Applied rewrites 62.4%

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

   if -2e53 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -1e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 3e4

   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 98.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -100000:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -100000:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2e+53)
     t_0
     (if (<= t_1 -100000.0) (* (/ x y) 4.0) (if (<= t_1 30000.0) 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 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (x / y) * 4.0;
	} else if (t_1 <= 30000.0) {
		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 <= (-2d+53)) then
        tmp = t_0
    else if (t_1 <= (-100000.0d0)) then
        tmp = (x / y) * 4.0d0
    else if (t_1 <= 30000.0d0) 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 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (x / y) * 4.0;
	} else if (t_1 <= 30000.0) {
		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 <= -2e+53:
		tmp = t_0
	elif t_1 <= -100000.0:
		tmp = (x / y) * 4.0
	elif t_1 <= 30000.0:
		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 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = Float64(Float64(x / y) * 4.0);
	elseif (t_1 <= 30000.0)
		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 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = (x / y) * 4.0;
	elseif (t_1 <= 30000.0)
		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, -2e+53], t$95$0, If[LessEqual[t$95$1, -100000.0], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[t$95$1, 30000.0], 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) < -2e53 or 3e4 < (/.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. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -2e53 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -1e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 3e4

   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 98.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -100000:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -100000:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2e+53)
     t_0
     (if (<= t_1 -100000.0) (* (/ 4.0 y) x) (if (<= t_1 30000.0) 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 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (4.0 / y) * x;
	} else if (t_1 <= 30000.0) {
		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 <= (-2d+53)) then
        tmp = t_0
    else if (t_1 <= (-100000.0d0)) then
        tmp = (4.0d0 / y) * x
    else if (t_1 <= 30000.0d0) 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 <= -2e+53) {
		tmp = t_0;
	} else if (t_1 <= -100000.0) {
		tmp = (4.0 / y) * x;
	} else if (t_1 <= 30000.0) {
		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 <= -2e+53:
		tmp = t_0
	elif t_1 <= -100000.0:
		tmp = (4.0 / y) * x
	elif t_1 <= 30000.0:
		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 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = Float64(Float64(4.0 / y) * x);
	elseif (t_1 <= 30000.0)
		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 <= -2e+53)
		tmp = t_0;
	elseif (t_1 <= -100000.0)
		tmp = (4.0 / y) * x;
	elseif (t_1 <= 30000.0)
		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, -2e+53], t$95$0, If[LessEqual[t$95$1, -100000.0], N[(N[(4.0 / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 30000.0], 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) < -2e53 or 3e4 < (/.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. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -2e53 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.3%

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

   if -1e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 3e4

   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 98.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -100000:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- x z) y) 4.0)) (t_1 (/ (* (- (+ (* 0.25 y) x) z) 4.0) y)))
   (if (<= t_1 -1e+21) t_0 (if (<= t_1 30000.0) (fma (/ x y) 4.0 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - z) / y) * 4.0;
	double t_1 = ((((0.25 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -1e+21) {
		tmp = t_0;
	} else if (t_1 <= 30000.0) {
		tmp = fma((x / y), 4.0, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - z) / y) * 4.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.25 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -1e+21)
		tmp = t_0;
	elseif (t_1 <= 30000.0)
		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[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $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, -1e+21], t$95$0, If[LessEqual[t$95$1, 30000.0], N[(N[(x / 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) < -1e21 or 3e4 < (/.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. *-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 99.6

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

   5. Applied rewrites 99.6%

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

   if -1e21 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 3e4

   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. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 30000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -100000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -100000.0) t_0 (if (<= t_1 30000.0) 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 <= -100000.0) {
		tmp = t_0;
	} else if (t_1 <= 30000.0) {
		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 <= (-100000.0d0)) then
        tmp = t_0
    else if (t_1 <= 30000.0d0) 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 <= -100000.0) {
		tmp = t_0;
	} else if (t_1 <= 30000.0) {
		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 <= -100000.0:
		tmp = t_0
	elif t_1 <= 30000.0:
		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 <= -100000.0)
		tmp = t_0;
	elseif (t_1 <= 30000.0)
		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 <= -100000.0)
		tmp = t_0;
	elseif (t_1 <= 30000.0)
		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, -100000.0], t$95$0, If[LessEqual[t$95$1, 30000.0], 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) < -1e5 or 3e4 < (/.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. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -1e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 3e4

   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 98.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -100000:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.25 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 30000:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 2.0)))
   (if (<= z -6.8e-15) t_0 (if (<= z 1.55e-58) (fma (/ x y) 4.0 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-4.0, (z / y), 2.0);
	double tmp;
	if (z <= -6.8e-15) {
		tmp = t_0;
	} else if (z <= 1.55e-58) {
		tmp = fma((x / y), 4.0, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-4.0, Float64(z / y), 2.0)
	tmp = 0.0
	if (z <= -6.8e-15)
		tmp = t_0;
	elseif (z <= 1.55e-58)
		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] + 2.0), $MachinePrecision]}, If[LessEqual[z, -6.8e-15], t$95$0, If[LessEqual[z, 1.55e-58], N[(N[(x / y), $MachinePrecision] * 4.0 + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000001e-15 or 1.55e-58 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-inverses N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. associate-*r/ N/A

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

      15. associate-*r* N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. metadata-eval N/A

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

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

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

   5. Applied rewrites 86.8%

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

   if -6.8000000000000001e-15 < z < 1.55e-58

   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. Applied rewrites 93.7%

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

   7. Applied rewrites 93.9%

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

Alternative 8: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 2.0)))
   (if (<= z -6.8e-15) t_0 (if (<= z 1.55e-58) (fma (/ 4.0 y) x 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-4.0, (z / y), 2.0);
	double tmp;
	if (z <= -6.8e-15) {
		tmp = t_0;
	} else if (z <= 1.55e-58) {
		tmp = fma((4.0 / y), x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-4.0, Float64(z / y), 2.0)
	tmp = 0.0
	if (z <= -6.8e-15)
		tmp = t_0;
	elseif (z <= 1.55e-58)
		tmp = fma(Float64(4.0 / y), x, 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[z, -6.8e-15], t$95$0, If[LessEqual[z, 1.55e-58], N[(N[(4.0 / y), $MachinePrecision] * x + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000001e-15 or 1.55e-58 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-inverses N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. associate-*r/ N/A

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

      15. associate-*r* N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. metadata-eval N/A

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

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

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

   5. Applied rewrites 86.8%

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

   if -6.8000000000000001e-15 < z < 1.55e-58

   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. Applied rewrites 93.7%

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

Alternative 9: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)))
   (if (<= x -7.2e+175) t_0 (if (<= x 1.35e+124) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * 4.0;
	double tmp;
	if (x <= -7.2e+175) {
		tmp = t_0;
	} else if (x <= 1.35e+124) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * 4.0)
	tmp = 0.0
	if (x <= -7.2e+175)
		tmp = t_0;
	elseif (x <= 1.35e+124)
		tmp = fma(-4.0, Float64(z / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+175], t$95$0, If[LessEqual[x, 1.35e+124], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000067e175 or 1.34999999999999989e124 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -7.20000000000000067e175 < x < 1.34999999999999989e124

   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-rgt-neg-out N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. associate-*r/ N/A

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

      15. associate-*r* N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. metadata-eval N/A

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

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

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

   5. Applied rewrites 87.9%

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

Alternative 10: 33.3% 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. Applied rewrites 36.4%

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

Reproduce

herbie shell --seed 2024285 
(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)))