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

Percentage Accurate: 99.7% → 100.0%

Time: 7.4s

Alternatives: 9

Speedup: 1.5×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

5. Applied rewrites 100.0%

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

Alternative 2: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -20000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (/ (* 4.0 x) y)))
   (if (<= t_1 -5e+90)
     t_0
     (if (<= t_1 -20000.0)
       t_2
       (if (<= t_1 10.0) 4.0 (if (<= t_1 5e+217) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -5e+90) {
		tmp = t_0;
	} else if (t_1 <= -20000.0) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+217) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = (z * (-4.0d0)) / y
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = (4.0d0 * x) / y
    if (t_1 <= (-5d+90)) then
        tmp = t_0
    else if (t_1 <= (-20000.0d0)) then
        tmp = t_2
    else if (t_1 <= 10.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 5d+217) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -5e+90) {
		tmp = t_0;
	} else if (t_1 <= -20000.0) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+217) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -4.0) / y
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = (4.0 * x) / y
	tmp = 0
	if t_1 <= -5e+90:
		tmp = t_0
	elif t_1 <= -20000.0:
		tmp = t_2
	elif t_1 <= 10.0:
		tmp = 4.0
	elif t_1 <= 5e+217:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (t_1 <= -5e+90)
		tmp = t_0;
	elseif (t_1 <= -20000.0)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+217)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -4.0) / y;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = (4.0 * x) / y;
	tmp = 0.0;
	if (t_1 <= -5e+90)
		tmp = t_0;
	elseif (t_1 <= -20000.0)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+217)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+90], t$95$0, If[LessEqual[t$95$1, -20000.0], t$95$2, If[LessEqual[t$95$1, 10.0], 4.0, If[LessEqual[t$95$1, 5e+217], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5.0000000000000004e90 or 10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5.00000000000000041e217

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if -5.0000000000000004e90 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 5.00000000000000041e217 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 10

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

Alternative 3: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -20000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (/ (* 4.0 x) y)))
   (if (<= t_1 -5e+90)
     t_0
     (if (<= t_1 -20000.0)
       t_2
       (if (<= t_1 10.0) 4.0 (if (<= t_1 5e+217) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -5e+90) {
		tmp = t_0;
	} else if (t_1 <= -20000.0) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+217) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = (4.0d0 * x) / y
    if (t_1 <= (-5d+90)) then
        tmp = t_0
    else if (t_1 <= (-20000.0d0)) then
        tmp = t_2
    else if (t_1 <= 10.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 5d+217) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -5e+90) {
		tmp = t_0;
	} else if (t_1 <= -20000.0) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else if (t_1 <= 5e+217) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = (4.0 * x) / y
	tmp = 0
	if t_1 <= -5e+90:
		tmp = t_0
	elif t_1 <= -20000.0:
		tmp = t_2
	elif t_1 <= 10.0:
		tmp = 4.0
	elif t_1 <= 5e+217:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (t_1 <= -5e+90)
		tmp = t_0;
	elseif (t_1 <= -20000.0)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+217)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = (4.0 * x) / y;
	tmp = 0.0;
	if (t_1 <= -5e+90)
		tmp = t_0;
	elseif (t_1 <= -20000.0)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	elseif (t_1 <= 5e+217)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+90], t$95$0, If[LessEqual[t$95$1, -20000.0], t$95$2, If[LessEqual[t$95$1, 10.0], 4.0, If[LessEqual[t$95$1, 5e+217], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5.0000000000000004e90 or 10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5.00000000000000041e217

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -5.0000000000000004e90 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 5.00000000000000041e217 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 10

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

Alternative 4: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - z\right)}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x z)) y)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -1e+18)
     t_0
     (if (<= t_1 2000000000000.0) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - z)) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = t_0;
	} else if (t_1 <= 2000000000000.0) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - z)) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = t_0;
	elseif (t_1 <= 2000000000000.0)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], t$95$0, If[LessEqual[t$95$1, 2000000000000.0], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e18 or 2e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -1e18 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2e12

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \color{blue}{3}\right) + 1 \]

      17. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \left(3 + 1\right)} \]

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.4%

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

Alternative 5: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -20000.0) t_0 (if (<= t_1 10.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 10.0d0) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 10.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 10.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 10.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 10.0], 4.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 10

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

Alternative 6: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ z y) -4.0 4.0)))
   (if (<= z -3.2e+56) t_0 (if (<= z 9e-19) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z / y), -4.0, 4.0);
	double tmp;
	if (z <= -3.2e+56) {
		tmp = t_0;
	} else if (z <= 9e-19) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z / y), -4.0, 4.0)
	tmp = 0.0
	if (z <= -3.2e+56)
		tmp = t_0;
	elseif (z <= 9e-19)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]}, If[LessEqual[z, -3.2e+56], t$95$0, If[LessEqual[z, 9e-19], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000003e56 or 9.00000000000000026e-19 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \color{blue}{3}\right) + 1 \]

      17. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \left(3 + 1\right)} \]

   5. Applied rewrites 88.8%

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

   7. Applied rewrites 88.9%

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

   if -3.20000000000000003e56 < z < 9.00000000000000026e-19

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{-1 \cdot x}{y}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{-1 \cdot x}{y}\right) \]

      7. *-inverses N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{-1 \cdot x}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{-1 \cdot x}{y}\right) \]

      9. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right) \]

      11. unsub-neg N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{x}{y}}\right) \]

      13. distribute-lft-in N/A

          \[\leadsto 1 + \color{blue}{\left(4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{y}\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + 4 \cdot \frac{3}{4}\right) + 4 \cdot \frac{x}{y}} \]

      15. metadata-eval N/A

          \[\leadsto \left(1 + \color{blue}{3}\right) + 4 \cdot \frac{x}{y} \]

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 90.2

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

   8. Applied rewrites 90.2%

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

Alternative 7: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (/ -4.0 y) 4.0)))
   (if (<= z -3.2e+56) t_0 (if (<= z 9e-19) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (-4.0 / y), 4.0);
	double tmp;
	if (z <= -3.2e+56) {
		tmp = t_0;
	} else if (z <= 9e-19) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(-4.0 / y), 4.0)
	tmp = 0.0
	if (z <= -3.2e+56)
		tmp = t_0;
	elseif (z <= 9e-19)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision] + 4.0), $MachinePrecision]}, If[LessEqual[z, -3.2e+56], t$95$0, If[LessEqual[z, 9e-19], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000003e56 or 9.00000000000000026e-19 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \color{blue}{3}\right) + 1 \]

      17. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \left(3 + 1\right)} \]

   5. Applied rewrites 88.8%

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

   if -3.20000000000000003e56 < z < 9.00000000000000026e-19

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{-1 \cdot x}{y}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{-1 \cdot x}{y}\right) \]

      7. *-inverses N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{-1 \cdot x}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{-1 \cdot x}{y}\right) \]

      9. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right) \]

      11. unsub-neg N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{x}{y}}\right) \]

      13. distribute-lft-in N/A

          \[\leadsto 1 + \color{blue}{\left(4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{y}\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + 4 \cdot \frac{3}{4}\right) + 4 \cdot \frac{x}{y}} \]

      15. metadata-eval N/A

          \[\leadsto \left(1 + \color{blue}{3}\right) + 4 \cdot \frac{x}{y} \]

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 90.2

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

   8. Applied rewrites 90.2%

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

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y)))
   (if (<= z -2.2e+110) t_0 (if (<= z 7.2e+179) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double tmp;
	if (z <= -2.2e+110) {
		tmp = t_0;
	} else if (z <= 7.2e+179) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	tmp = 0.0
	if (z <= -2.2e+110)
		tmp = t_0;
	elseif (z <= 7.2e+179)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -2.2e+110], t$95$0, If[LessEqual[z, 7.2e+179], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.19999999999999992e110 or 7.1999999999999995e179 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.1

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

   8. Applied rewrites 86.1%

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

   if -2.19999999999999992e110 < z < 7.1999999999999995e179

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{-1 \cdot x}{y}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{-1 \cdot x}{y}\right) \]

      7. *-inverses N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{-1 \cdot x}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{-1 \cdot x}{y}\right) \]

      9. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right) \]

      11. unsub-neg N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{x}{y}}\right) \]

      13. distribute-lft-in N/A

          \[\leadsto 1 + \color{blue}{\left(4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{y}\right)} \]

      14. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + 4 \cdot \frac{3}{4}\right) + 4 \cdot \frac{x}{y}} \]

      15. metadata-eval N/A

          \[\leadsto \left(1 + \color{blue}{3}\right) + 4 \cdot \frac{x}{y} \]

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 82.8

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

   8. Applied rewrites 82.8%

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

Alternative 9: 34.3% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.5%

   \[\leadsto \color{blue}{4} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))