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

Percentage Accurate: 99.9% → 100.0%

Time: 7.1s

Alternatives: 7

Speedup: 1.1×

• 21.0% 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: 99.9% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 100.0

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

   7. lift--.f64 N/A

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

   8. lift--.f64 N/A

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

   9. associate--l- N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{z} \cdot 4} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - y\right)}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x y)) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -5e+28) t_0 (if (<= t_1 10000.0) (fma 4.0 (/ x z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - y)) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -5e+28) {
		tmp = t_0;
	} else if (t_1 <= 10000.0) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - y)) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -5e+28)
		tmp = t_0;
	elseif (t_1 <= 10000.0)
		tmp = fma(4.0, Float64(x / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+28], t$95$0, If[LessEqual[t$95$1, 10000.0], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4.99999999999999957e28 or 1e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 98.2%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if -4.99999999999999957e28 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1e4

   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. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 3: 66.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_0 -5e+28)
     (/ (* y -4.0) z)
     (if (<= t_0 -1.0) -2.0 (/ (* x 4.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = (x * 4.0) / 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) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_0 <= (-5d+28)) then
        tmp = (y * (-4.0d0)) / z
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = (x * 4.0d0) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = (x * 4.0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_0 <= -5e+28:
		tmp = (y * -4.0) / z
	elif t_0 <= -1.0:
		tmp = -2.0
	else:
		tmp = (x * 4.0) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_0 <= -5e+28)
		tmp = Float64(Float64(y * -4.0) / z);
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = Float64(Float64(x * 4.0) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_0 <= -5e+28)
		tmp = (y * -4.0) / z;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = (x * 4.0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+28], N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, -1.0], -2.0, N[(N[(x * 4.0), $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) < -4.99999999999999957e28

   1. Initial program 96.2%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 54.6

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

   5. Applied rewrites 54.6%

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

   if -4.99999999999999957e28 < (/.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 90.3%

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

   if -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 x around inf

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

      1. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y -4.0) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -5e+28) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * -4.0) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -5e+28) {
		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 = (y * (-4.0d0)) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-5d+28)) 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 = (y * -4.0) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -5e+28) {
		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 = (y * -4.0) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -5e+28:
		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(y * -4.0) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -5e+28)
		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 = (y * -4.0) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -5e+28)
		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[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+28], 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) < -4.99999999999999957e28 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 98.2%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   if -4.99999999999999957e28 < (/.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 90.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 4.0 (/ x z) -2.0)))
   (if (<= x -6.2e+28) t_0 (if (<= x 1.3e+134) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(4.0, (x / z), -2.0);
	double tmp;
	if (x <= -6.2e+28) {
		tmp = t_0;
	} else if (x <= 1.3e+134) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(4.0, Float64(x / z), -2.0)
	tmp = 0.0
	if (x <= -6.2e+28)
		tmp = t_0;
	elseif (x <= 1.3e+134)
		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[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[x, -6.2e+28], t$95$0, If[LessEqual[x, 1.3e+134], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2000000000000001e28 or 1.3000000000000001e134 < x

   1. Initial program 98.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -6.2000000000000001e28 < x < 1.3000000000000001e134

   1. Initial program 98.8%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 89.3%

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

   7. Applied rewrites 89.4%

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

Alternative 6: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -3.4e+160) t_0 (if (<= x 3.1e+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 <= -3.4e+160) {
		tmp = t_0;
	} else if (x <= 3.1e+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 <= -3.4e+160)
		tmp = t_0;
	elseif (x <= 3.1e+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, -3.4e+160], t$95$0, If[LessEqual[x, 3.1e+135], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000003e160 or 3.10000000000000022e135 < 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. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -3.4000000000000003e160 < x < 3.10000000000000022e135

   1. Initial program 98.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 85.2%

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

   7. Applied rewrites 85.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+160}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.7% 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 98.8%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 35.1%

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

Developer Target 1: 98.0% 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 2024220 
(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))