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

Percentage Accurate: 99.7% → 99.7%

Time: 6.4s

Alternatives: 8

Speedup: 1.0×

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

Specification

Math

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

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: 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]

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. Add Preprocessing

Alternative 2: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_1 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000000000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)) (t_1 (/ (* 4.0 x) y)))
   (if (<= t_0 -1e+97)
     (/ (* z -4.0) y)
     (if (<= t_0 -50.0) t_1 (if (<= t_0 50000000000.0) 4.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = (z * -4.0) / y;
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_1 = (4.0d0 * x) / y
    if (t_0 <= (-1d+97)) then
        tmp = (z * (-4.0d0)) / y
    else if (t_0 <= (-50.0d0)) then
        tmp = t_1
    else if (t_0 <= 50000000000.0d0) then
        tmp = 4.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = (z * -4.0) / y;
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_1 = (4.0 * x) / y
	tmp = 0
	if t_0 <= -1e+97:
		tmp = (z * -4.0) / y
	elif t_0 <= -50.0:
		tmp = t_1
	elif t_0 <= 50000000000.0:
		tmp = 4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_1 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (t_0 <= -1e+97)
		tmp = Float64(Float64(z * -4.0) / y);
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_1 = (4.0 * x) / y;
	tmp = 0.0;
	if (t_0 <= -1e+97)
		tmp = (z * -4.0) / y;
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+97], N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 50000000000.0], 4.0, t$95$1]]]]]

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) < -1.0000000000000001e97

   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 62.3

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

   5. Simplified 62.3%

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 62.4

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

   7. Applied egg-rr 62.4%

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

   if -1.0000000000000001e97 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -50 or 5e10 < (/.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 56.9

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

   5. Simplified 56.9%

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

   if -50 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e10

   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. Simplified 92.3%

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

Alternative 3: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_1 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000000000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)) (t_1 (/ (* 4.0 x) y)))
   (if (<= t_0 -1e+97)
     (* z (/ -4.0 y))
     (if (<= t_0 -50.0) t_1 (if (<= t_0 50000000000.0) 4.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = z * (-4.0 / y);
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_1 = (4.0d0 * x) / y
    if (t_0 <= (-1d+97)) then
        tmp = z * ((-4.0d0) / y)
    else if (t_0 <= (-50.0d0)) then
        tmp = t_1
    else if (t_0 <= 50000000000.0d0) then
        tmp = 4.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (t_0 <= -1e+97) {
		tmp = z * (-4.0 / y);
	} else if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000000.0) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_1 = (4.0 * x) / y
	tmp = 0
	if t_0 <= -1e+97:
		tmp = z * (-4.0 / y)
	elif t_0 <= -50.0:
		tmp = t_1
	elif t_0 <= 50000000000.0:
		tmp = 4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_1 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (t_0 <= -1e+97)
		tmp = Float64(z * Float64(-4.0 / y));
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_1 = (4.0 * x) / y;
	tmp = 0.0;
	if (t_0 <= -1e+97)
		tmp = z * (-4.0 / y);
	elseif (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 50000000000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+97], N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 50000000000.0], 4.0, t$95$1]]]]]

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) < -1.0000000000000001e97

   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 62.3

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

   5. Simplified 62.3%

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

   if -1.0000000000000001e97 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -50 or 5e10 < (/.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 56.9

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

   5. Simplified 56.9%

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

   if -50 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e10

   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. Simplified 92.3%

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

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - z\right)}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5e+18)
     t_0
     (if (<= t_1 50000000000.0) (fma z (/ -4.0 y) 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 <= -5e+18) {
		tmp = t_0;
	} else if (t_1 <= 50000000000.0) {
		tmp = fma(z, (-4.0 / y), 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 <= -5e+18)
		tmp = t_0;
	elseif (t_1 <= 50000000000.0)
		tmp = fma(z, Float64(-4.0 / y), 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, -5e+18], t$95$0, If[LessEqual[t$95$1, 50000000000.0], N[(z * N[(-4.0 / y), $MachinePrecision] + 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) < -5e18 or 5e10 < (/.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. Simplified 99.9%

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

   if -5e18 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e10

   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. Simplified 97.5%

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

Alternative 5: 66.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -50.0) t_0 (if (<= t_1 5.0) 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 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= (-50.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) 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 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= -50.0:
		tmp = t_0
	elif t_1 <= 5.0:
		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 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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, -50.0], t$95$0, If[LessEqual[t$95$1, 5.0], 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) < -50 or 5 < (/.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 53.2

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

   5. Simplified 53.2%

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

   if -50 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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. Simplified 93.2%

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

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (/ -4.0 y) 4.0)))
   (if (<= z -2.8e+47) t_0 (if (<= z 3.8e-13) (fma (/ x y) 4.0 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 <= -2.8e+47) {
		tmp = t_0;
	} else if (z <= 3.8e-13) {
		tmp = fma((x / y), 4.0, 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 <= -2.8e+47)
		tmp = t_0;
	elseif (z <= 3.8e-13)
		tmp = fma(Float64(x / y), 4.0, 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, -2.8e+47], t$95$0, If[LessEqual[z, 3.8e-13], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999988e47 or 3.8e-13 < 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. Simplified 86.0%

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

   if -2.79999999999999988e47 < z < 3.8e-13

   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 z around 0

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

   5. Simplified 93.6%

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

      1. lift-/.f64 N/A

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

      2. distribute-neg-in N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. associate-*l/ N/A

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

      18. *-lft-identity N/A

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

      19. lower-/.f64 93.8

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

   7. Applied egg-rr 93.8%

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

Alternative 7: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-4}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) y)))
   (if (<= x -4.5e+176) t_0 (if (<= x 7e+170) (fma z (/ -4.0 y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / y;
	double tmp;
	if (x <= -4.5e+176) {
		tmp = t_0;
	} else if (x <= 7e+170) {
		tmp = fma(z, (-4.0 / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (x <= -4.5e+176)
		tmp = t_0;
	elseif (x <= 7e+170)
		tmp = fma(z, Float64(-4.0 / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -4.5e+176], t$95$0, If[LessEqual[x, 7e+170], N[(z * N[(-4.0 / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000003e176 or 7.00000000000000011e170 < x

   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 84.6

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

   5. Simplified 84.6%

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

   if -4.50000000000000003e176 < x < 7.00000000000000011e170

   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

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   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. Simplified 82.6%

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

Alternative 8: 33.8% 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. Simplified 34.7%

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

Reproduce

herbie shell --seed 2024207 
(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)))