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

Percentage Accurate: 99.8% → 100.0%

Time: 6.4s

Alternatives: 8

Speedup: 1.5×

• 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: 99.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. *-inverses N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+236}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+14} \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_0 -5e+236)
     (* (/ z y) -4.0)
     (if (or (<= t_0 -1e+14) (not (<= t_0 500.0))) (* (/ x y) 4.0) 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_0 <= -5e+236) {
		tmp = (z / y) * -4.0;
	} else if ((t_0 <= -1e+14) || !(t_0 <= 500.0)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = 4.0;
	}
	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) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_0 <= (-5d+236)) then
        tmp = (z / y) * (-4.0d0)
    else if ((t_0 <= (-1d+14)) .or. (.not. (t_0 <= 500.0d0))) then
        tmp = (x / y) * 4.0d0
    else
        tmp = 4.0d0
    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 tmp;
	if (t_0 <= -5e+236) {
		tmp = (z / y) * -4.0;
	} else if ((t_0 <= -1e+14) || !(t_0 <= 500.0)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_0 <= -5e+236:
		tmp = (z / y) * -4.0
	elif (t_0 <= -1e+14) or not (t_0 <= 500.0):
		tmp = (x / y) * 4.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_0 <= -5e+236)
		tmp = Float64(Float64(z / y) * -4.0);
	elseif ((t_0 <= -1e+14) || !(t_0 <= 500.0))
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_0 <= -5e+236)
		tmp = (z / y) * -4.0;
	elseif ((t_0 <= -1e+14) || ~((t_0 <= 500.0)))
		tmp = (x / y) * 4.0;
	else
		tmp = 4.0;
	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]}, If[LessEqual[t$95$0, -5e+236], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], If[Or[LessEqual[t$95$0, -1e+14], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], 4.0]]]

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) < -4.9999999999999997e236

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.8%

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

   if -4.9999999999999997e236 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -1e14 or 500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

   if -1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 500

   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 94.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -5 \cdot 10^{+236}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1 \cdot 10^{+14} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 500\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14} \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -1e+14) (not (<= t_0 500.0)))
     (* (/ (- x z) y) 4.0)
     (fma (/ z y) -4.0 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -1e+14) || !(t_0 <= 500.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma((z / y), -4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -1e+14) || !(t_0 <= 500.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(Float64(z / y), -4.0, 4.0);
	end
	return 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]}, If[Or[LessEqual[t$95$0, -1e+14], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]]]

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) < -1e14 or 500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 500

   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 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1 \cdot 10^{+14} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 500\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -1e+14) (not (<= t_0 4e+20))) (* (/ z y) -4.0) 4.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -1e+14) || !(t_0 <= 4e+20)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 4.0;
	}
	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) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if ((t_0 <= (-1d+14)) .or. (.not. (t_0 <= 4d+20))) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = 4.0d0
    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 tmp;
	if ((t_0 <= -1e+14) || !(t_0 <= 4e+20)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if (t_0 <= -1e+14) or not (t_0 <= 4e+20):
		tmp = (z / y) * -4.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -1e+14) || !(t_0 <= 4e+20))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if ((t_0 <= -1e+14) || ~((t_0 <= 4e+20)))
		tmp = (z / y) * -4.0;
	else
		tmp = 4.0;
	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]}, If[Or[LessEqual[t$95$0, -1e+14], N[Not[LessEqual[t$95$0, 4e+20]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 4.0]]

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) < -1e14 or 4e20 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.6%

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

   if -1e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 4e20

   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 92.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1 \cdot 10^{+14} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -126000 \lor \neg \left(x \leq 44000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -126000.0) (not (<= x 44000000000000.0)))
   (fma (/ 4.0 y) x 4.0)
   (fma (/ z y) -4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -126000.0) || !(x <= 44000000000000.0)) {
		tmp = fma((4.0 / y), x, 4.0);
	} else {
		tmp = fma((z / y), -4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -126000.0) || !(x <= 44000000000000.0))
		tmp = fma(Float64(4.0 / y), x, 4.0);
	else
		tmp = fma(Float64(z / y), -4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -126000.0], N[Not[LessEqual[x, 44000000000000.0]], $MachinePrecision]], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -126000 or 4.4e13 < 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 z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -126000 < x < 4.4e13

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -126000 \lor \neg \left(x \leq 44000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+145} \lor \neg \left(x \leq 7.5 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.8e+145) (not (<= x 7.5e+80)))
   (* (/ x y) 4.0)
   (fma (/ z y) -4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.8e+145) || !(x <= 7.5e+80)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = fma((z / y), -4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.8e+145) || !(x <= 7.5e+80))
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = fma(Float64(z / y), -4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.8e+145], N[Not[LessEqual[x, 7.5e+80]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.7999999999999995e145 or 7.49999999999999994e80 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -7.7999999999999995e145 < x < 7.49999999999999994e80

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+145} \lor \neg \left(x \leq 7.5 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. *-inverses N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. div-add-rev N/A

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

   4. associate--l+ N/A

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

   5. div-sub N/A

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

   6. distribute-lft-out-- N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. +-commutative N/A

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

   10. +-commutative N/A

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

   11. div-add N/A

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

   12. associate-/l* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

8. Applied rewrites 99.8%

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

Alternative 8: 34.8% 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 99.5%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 32.9%

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

Reproduce

herbie shell --seed 2024329 
(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)))