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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 7

Speedup: 1.0×

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

Specification

Math

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

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: 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]

Derivation

1. Initial program 99.6%

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

Alternative 2: 67.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ t_2 := \frac{y \cdot -4}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) z))
        (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z))
        (t_2 (/ (* y -4.0) z)))
   (if (<= t_1 -5e+264)
     t_0
     (if (<= t_1 -4e+122)
       t_2
       (if (<= t_1 -5000.0)
         t_0
         (if (<= t_1 -1.0) -2.0 (if (<= t_1 1e+152) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (y * -4.0) / z;
	double tmp;
	if (t_1 <= -5e+264) {
		tmp = t_0;
	} else if (t_1 <= -4e+122) {
		tmp = t_2;
	} else if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 1e+152) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (4.0d0 * x) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    t_2 = (y * (-4.0d0)) / z
    if (t_1 <= (-5d+264)) then
        tmp = t_0
    else if (t_1 <= (-4d+122)) then
        tmp = t_2
    else if (t_1 <= (-5000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 1d+152) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (y * -4.0) / z;
	double tmp;
	if (t_1 <= -5e+264) {
		tmp = t_0;
	} else if (t_1 <= -4e+122) {
		tmp = t_2;
	} else if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 1e+152) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * x) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	t_2 = (y * -4.0) / z
	tmp = 0
	if t_1 <= -5e+264:
		tmp = t_0
	elif t_1 <= -4e+122:
		tmp = t_2
	elif t_1 <= -5000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 1e+152:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	t_2 = Float64(Float64(y * -4.0) / z)
	tmp = 0.0
	if (t_1 <= -5e+264)
		tmp = t_0;
	elseif (t_1 <= -4e+122)
		tmp = t_2;
	elseif (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 1e+152)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * x) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	t_2 = (y * -4.0) / z;
	tmp = 0.0;
	if (t_1 <= -5e+264)
		tmp = t_0;
	elseif (t_1 <= -4e+122)
		tmp = t_2;
	elseif (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 1e+152)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $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]}, Block[{t$95$2 = N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+264], t$95$0, If[LessEqual[t$95$1, -4e+122], t$95$2, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 1e+152], t$95$2, t$95$0]]]]]]]]

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) < -5.00000000000000033e264 or -4.00000000000000006e122 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5e3 or 1e152 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.1%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 61.7

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

   5. Applied rewrites 61.7%

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

   if -5.00000000000000033e264 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4.00000000000000006e122 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1e152

   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 69.3

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

   5. Applied rewrites 69.3%

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

   if -5e3 < (/.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%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -4 \cdot 10^{+122}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5000:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 10^{+152}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - y\right)}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 40:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1e+16) t_0 (if (<= t_1 40.0) (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 <= -1e+16) {
		tmp = t_0;
	} else if (t_1 <= 40.0) {
		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 <= -1e+16)
		tmp = t_0;
	elseif (t_1 <= 40.0)
		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, -1e+16], t$95$0, If[LessEqual[t$95$1, 40.0], 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) < -1e16 or 40 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 40

   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 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 4: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1e+16) 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 <= -1e+16) {
		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 <= (-1d+16)) 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 <= -1e+16) {
		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 <= -1e+16:
		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 <= -1e+16)
		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 <= -1e+16)
		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, -1e+16], 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) < -1e16 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 51.3

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

   5. Applied rewrites 51.3%

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

   if -1e16 < (/.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 95.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ -4.0 z) y -2.0)))
   (if (<= y -1.45e+102) t_0 (if (<= y 7.5e+151) (fma 4.0 (/ x z) -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 (y <= -1.45e+102) {
		tmp = t_0;
	} else if (y <= 7.5e+151) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-4.0 / z), y, -2.0)
	tmp = 0.0
	if (y <= -1.45e+102)
		tmp = t_0;
	elseif (y <= 7.5e+151)
		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 / z), $MachinePrecision] * y + -2.0), $MachinePrecision]}, If[LessEqual[y, -1.45e+102], t$95$0, If[LessEqual[y, 7.5e+151], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4500000000000001e102 or 7.49999999999999977e151 < y

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 90.4%

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

   7. Applied rewrites 90.4%

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

   if -1.4500000000000001e102 < y < 7.49999999999999977e151

   1. Initial program 99.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

Alternative 6: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y -4.0) z)))
   (if (<= y -1.95e+111) t_0 (if (<= y 2.1e+225) (fma 4.0 (/ x z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * -4.0) / z;
	double tmp;
	if (y <= -1.95e+111) {
		tmp = t_0;
	} else if (y <= 2.1e+225) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * -4.0) / z)
	tmp = 0.0
	if (y <= -1.95e+111)
		tmp = t_0;
	elseif (y <= 2.1e+225)
		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[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.95e+111], t$95$0, If[LessEqual[y, 2.1e+225], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9499999999999999e111 or 2.1e225 < y

   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 85.1

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

   5. Applied rewrites 85.1%

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

   if -1.9499999999999999e111 < y < 2.1e225

   1. Initial program 99.5%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+111}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% 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 99.6%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 37.7%

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

Developer Target 1: 97.5% 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 2024221 
(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))