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

Percentage Accurate: 99.8% → 100.0%

Time: 8.6s

Alternatives: 11

Speedup: 1.3×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.9

      \[\leadsto \frac{4 \cdot \left(\frac{1}{\color{blue}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

4. Applied rewrites 99.9%

   \[\leadsto \frac{4 \cdot \left(\color{blue}{\frac{1}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

5. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -10000000.0)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 2e+104) t_0 (/ (* -4.0 y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	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) / z
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-10000000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+104) then
        tmp = t_0
    else
        tmp = ((-4.0d0) * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / z
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+104:
		tmp = t_0
	else:
		tmp = (-4.0 * y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = Float64(Float64(-4.0 * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / z;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = (-4.0 * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+104], t$95$0, N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e7 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e104

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if -1e7 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 96.8%

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

   if 2e104 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -10000000:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{z} \cdot x\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 z) x)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -10000000.0)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 2e+104) t_0 (/ (* -4.0 y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	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) * x
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-10000000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+104) then
        tmp = t_0
    else
        tmp = ((-4.0d0) * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / z) * x
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+104:
		tmp = t_0
	else:
		tmp = (-4.0 * y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / z) * x)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = Float64(Float64(-4.0 * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / z) * x;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = (-4.0 * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+104], t$95$0, N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e7 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e104

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if -1e7 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 96.8%

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

   if 2e104 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -10000000:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{z} \cdot x\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 z) x)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -10000000.0)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 2e+104) t_0 (* (/ -4.0 z) y))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 / z) * y;
	}
	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) * x
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-10000000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+104) then
        tmp = t_0
    else
        tmp = ((-4.0d0) / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+104) {
		tmp = t_0;
	} else {
		tmp = (-4.0 / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / z) * x
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+104:
		tmp = t_0
	else:
		tmp = (-4.0 / z) * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / z) * x)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = Float64(Float64(-4.0 / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / z) * x;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+104)
		tmp = t_0;
	else
		tmp = (-4.0 / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+104], t$95$0, N[(N[(-4.0 / z), $MachinePrecision] * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e7 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e104

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if -1e7 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 96.8%

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

   if 2e104 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 59.1

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

   5. Applied rewrites 59.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -10000000:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) 4.0) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -2e+31) t_0 (if (<= t_1 1e+14) (fma (/ x z) 4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - y) * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_0;
	} else if (t_1 <= 1e+14) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - y) * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -2e+31)
		tmp = t_0;
	elseif (t_1 <= 1e+14)
		tmp = fma(Float64(x / z), 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 - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+31], t$95$0, If[LessEqual[t$95$1, 1e+14], N[(N[(x / z), $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 y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1.9999999999999999e31 or 1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.9999999999999999e31 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1e14

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 100.0

         \[\leadsto \frac{4 \cdot \left(\frac{1}{\color{blue}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{4 \cdot \left(\color{blue}{\frac{1}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-frac-neg N/A

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

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

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z}} \]

      7. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z} \]

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{z} \cdot y\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 z) y)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -2e+31) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / z) * y;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		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 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-2d+31)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) 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 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -2e+31) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 / z) * y
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -2e+31:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 / z) * y)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -2e+31)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		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 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -2e+31)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		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 / z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+31], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1.9999999999999999e31 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 48.9

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

   5. Applied rewrites 48.9%

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

   if -1.9999999999999999e31 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 92.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 4.0 -2.0)))
   (if (<= x -3.6e+85) t_0 (if (<= x 2.5e-73) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 4.0, -2.0);
	double tmp;
	if (x <= -3.6e+85) {
		tmp = t_0;
	} else if (x <= 2.5e-73) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 4.0, -2.0)
	tmp = 0.0
	if (x <= -3.6e+85)
		tmp = t_0;
	elseif (x <= 2.5e-73)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]}, If[LessEqual[x, -3.6e+85], t$95$0, If[LessEqual[x, 2.5e-73], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999998e85 or 2.4999999999999999e-73 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.9

         \[\leadsto \frac{4 \cdot \left(\frac{1}{\color{blue}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{4 \cdot \left(\color{blue}{\frac{1}{\frac{1}{x - y}}} - z \cdot 0.5\right)}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-frac-neg N/A

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

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

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z}} \]

      7. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \frac{\frac{-1}{2} \cdot z}{z} \]

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 85.9

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 86.1%

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

   if -3.5999999999999998e85 < x < 2.4999999999999999e-73

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]

   5. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\left(-0.5 - \frac{y}{z}\right) \cdot 4} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.5%

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

Alternative 8: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 4.0 z) x -2.0)))
   (if (<= x -3.6e+85) t_0 (if (<= x 2.5e-73) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((4.0 / z), x, -2.0);
	double tmp;
	if (x <= -3.6e+85) {
		tmp = t_0;
	} else if (x <= 2.5e-73) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(4.0 / z), x, -2.0)
	tmp = 0.0
	if (x <= -3.6e+85)
		tmp = t_0;
	elseif (x <= 2.5e-73)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]}, If[LessEqual[x, -3.6e+85], t$95$0, If[LessEqual[x, 2.5e-73], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999998e85 or 2.4999999999999999e-73 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -3.5999999999999998e85 < x < 2.4999999999999999e-73

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]

   5. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\left(-0.5 - \frac{y}{z}\right) \cdot 4} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.5%

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

Alternative 9: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -8.8e+145) t_0 (if (<= x 4.6e+135) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double tmp;
	if (x <= -8.8e+145) {
		tmp = t_0;
	} else if (x <= 4.6e+135) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	tmp = 0.0
	if (x <= -8.8e+145)
		tmp = t_0;
	elseif (x <= 4.6e+135)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -8.8e+145], t$95$0, If[LessEqual[x, 4.6e+135], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.80000000000000035e145 or 4.6000000000000002e135 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -8.80000000000000035e145 < x < 4.6000000000000002e135

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\left(-0.5 - \frac{y}{z}\right) \cdot 4} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.5%

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

Alternative 10: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-fma.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 11: 33.9% accurate, 28.0× 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%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

5. Applied rewrites 35.6%

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

Developer Target 1: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))