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

Percentage Accurate: 99.9% → 100.0%

Time: 5.1s

Alternatives: 9

Speedup: 1.4×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{x - z}{y \cdot 0.25} + 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (/ (- x z) (* y 0.25)) 2.0))

C

double code(double x, double y, double z) {
	return ((x - z) / (y * 0.25)) + 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - z) / (y * 0.25d0)) + 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return ((x - z) / (y * 0.25)) + 2.0;
}

Python

def code(x, y, z):
	return ((x - z) / (y * 0.25)) + 2.0

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - z) / Float64(y * 0.25)) + 2.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - z) / (y * 0.25)) + 2.0;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - z), $MachinePrecision] / N[(y * 0.25), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} + 1} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} + 1 \]

   3. +-commutative 99.7%

      \[\leadsto \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) + 1 \]

   4. associate--l+ 99.7%

      \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} + 1 \]

   5. +-commutative 99.7%

      \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} + 1 \]

   6. distribute-lft-in 99.8%

      \[\leadsto \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} + 1 \]

   7. associate-+l+ 99.8%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + \left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + 1\right)} \]

   8. associate-*l/ 99.8%

      \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\color{blue}{\frac{4 \cdot \left(y \cdot 0.25\right)}{y}} + 1\right) \]

   9. *-commutative 99.8%

      \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{\left(y \cdot 0.25\right) \cdot 4}}{y} + 1\right) \]

   10. associate-*l* 99.8%

       \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{y \cdot \left(0.25 \cdot 4\right)}}{y} + 1\right) \]

   11. metadata-eval 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. *-inverses 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + 2} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

      \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

   4. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

   5. *-un-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]
7. Add Preprocessing

Alternative 2: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+66} \lor \neg \left(z \leq -7.4 \cdot 10^{-6}\right) \land \left(z \leq -3.4 \cdot 10^{-10} \lor \neg \left(z \leq 7.2 \cdot 10^{+28}\right)\right):\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.5e+66)
         (and (not (<= z -7.4e-6)) (or (<= z -3.4e-10) (not (<= z 7.2e+28)))))
   (+ 2.0 (* -4.0 (/ z y)))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e+66) || (!(z <= -7.4e-6) && ((z <= -3.4e-10) || !(z <= 7.2e+28)))) {
		tmp = 2.0 + (-4.0 * (z / y));
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	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) :: tmp
    if ((z <= (-9.5d+66)) .or. (.not. (z <= (-7.4d-6))) .and. (z <= (-3.4d-10)) .or. (.not. (z <= 7.2d+28))) then
        tmp = 2.0d0 + ((-4.0d0) * (z / y))
    else
        tmp = 2.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e+66) || (!(z <= -7.4e-6) && ((z <= -3.4e-10) || !(z <= 7.2e+28)))) {
		tmp = 2.0 + (-4.0 * (z / y));
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.5e+66) or (not (z <= -7.4e-6) and ((z <= -3.4e-10) or not (z <= 7.2e+28))):
		tmp = 2.0 + (-4.0 * (z / y))
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.5e+66) || (!(z <= -7.4e-6) && ((z <= -3.4e-10) || !(z <= 7.2e+28))))
		tmp = Float64(2.0 + Float64(-4.0 * Float64(z / y)));
	else
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.5e+66) || (~((z <= -7.4e-6)) && ((z <= -3.4e-10) || ~((z <= 7.2e+28)))))
		tmp = 2.0 + (-4.0 * (z / y));
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.5e+66], And[N[Not[LessEqual[z, -7.4e-6]], $MachinePrecision], Or[LessEqual[z, -3.4e-10], N[Not[LessEqual[z, 7.2e+28]], $MachinePrecision]]]], N[(2.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.50000000000000051e66 or -7.4000000000000003e-6 < z < -3.40000000000000015e-10 or 7.1999999999999999e28 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} + 1} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} + 1 \]

      3. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) + 1 \]

      4. associate--l+ 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} + 1 \]

      5. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} + 1 \]

      6. distribute-lft-in 99.8%

         \[\leadsto \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} + 1 \]

      7. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + \left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + 1\right)} \]

      8. associate-*l/ 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\color{blue}{\frac{4 \cdot \left(y \cdot 0.25\right)}{y}} + 1\right) \]

      9. *-commutative 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{\left(y \cdot 0.25\right) \cdot 4}}{y} + 1\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{y \cdot \left(0.25 \cdot 4\right)}}{y} + 1\right) \]

      11. metadata-eval 99.8%

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

      12. *-rgt-identity 99.8%

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

      13. *-inverses 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + 2} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.8%

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

      2. div-inv 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around 0 85.5%

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

   if -9.50000000000000051e66 < z < -7.4000000000000003e-6 or -3.40000000000000015e-10 < z < 7.1999999999999999e28

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} + 1} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} + 1 \]

      3. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) + 1 \]

      4. associate--l+ 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} + 1 \]

      5. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} + 1 \]

      6. distribute-lft-in 99.8%

         \[\leadsto \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} + 1 \]

      7. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + \left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + 1\right)} \]

      8. associate-*l/ 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\color{blue}{\frac{4 \cdot \left(y \cdot 0.25\right)}{y}} + 1\right) \]

      9. *-commutative 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{\left(y \cdot 0.25\right) \cdot 4}}{y} + 1\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{y \cdot \left(0.25 \cdot 4\right)}}{y} + 1\right) \]

      11. metadata-eval 99.8%

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

      12. *-rgt-identity 99.8%

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

      13. *-inverses 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + 2} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.8%

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

      2. div-inv 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around inf 88.7%

      \[\leadsto \color{blue}{4 \cdot \frac{x}{y}} + 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+66} \lor \neg \left(z \leq -7.4 \cdot 10^{-6}\right) \land \left(z \leq -3.4 \cdot 10^{-10} \lor \neg \left(z \leq 7.2 \cdot 10^{+28}\right)\right):\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+161}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+83} \lor \neg \left(y \leq -2.9 \cdot 10^{+59}\right) \land y \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e+161)
   2.0
   (if (or (<= y -6.5e+83) (and (not (<= y -2.9e+59)) (<= y 1.05e-30)))
     (+ (* -4.0 (/ z y)) 1.0)
     2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+161) {
		tmp = 2.0;
	} else if ((y <= -6.5e+83) || (!(y <= -2.9e+59) && (y <= 1.05e-30))) {
		tmp = (-4.0 * (z / y)) + 1.0;
	} else {
		tmp = 2.0;
	}
	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) :: tmp
    if (y <= (-1.55d+161)) then
        tmp = 2.0d0
    else if ((y <= (-6.5d+83)) .or. (.not. (y <= (-2.9d+59))) .and. (y <= 1.05d-30)) then
        tmp = ((-4.0d0) * (z / y)) + 1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+161) {
		tmp = 2.0;
	} else if ((y <= -6.5e+83) || (!(y <= -2.9e+59) && (y <= 1.05e-30))) {
		tmp = (-4.0 * (z / y)) + 1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e+161:
		tmp = 2.0
	elif (y <= -6.5e+83) or (not (y <= -2.9e+59) and (y <= 1.05e-30)):
		tmp = (-4.0 * (z / y)) + 1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e+161)
		tmp = 2.0;
	elseif ((y <= -6.5e+83) || (!(y <= -2.9e+59) && (y <= 1.05e-30)))
		tmp = Float64(Float64(-4.0 * Float64(z / y)) + 1.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e+161)
		tmp = 2.0;
	elseif ((y <= -6.5e+83) || (~((y <= -2.9e+59)) && (y <= 1.05e-30)))
		tmp = (-4.0 * (z / y)) + 1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e+161], 2.0, If[Or[LessEqual[y, -6.5e+83], And[N[Not[LessEqual[y, -2.9e+59]], $MachinePrecision], LessEqual[y, 1.05e-30]]], N[(N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000003e161 or -6.5000000000000003e83 < y < -2.89999999999999991e59 or 1.0500000000000001e-30 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.8%

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

   if -1.55000000000000003e161 < y < -6.5000000000000003e83 or -2.89999999999999991e59 < y < 1.0500000000000001e-30

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 51.7%

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

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

      \[\leadsto 1 + \color{blue}{\frac{z}{y} \cdot -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+161}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+83} \lor \neg \left(y \leq -2.9 \cdot 10^{+59}\right) \land y \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+161}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{+87} \lor \neg \left(y \leq -4.5 \cdot 10^{+60}\right) \land y \leq 10^{-30}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+161)
   2.0
   (if (or (<= y -1.22e+87) (and (not (<= y -4.5e+60)) (<= y 1e-30)))
     (+ 1.0 (* z (/ -4.0 y)))
     2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+161) {
		tmp = 2.0;
	} else if ((y <= -1.22e+87) || (!(y <= -4.5e+60) && (y <= 1e-30))) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = 2.0;
	}
	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) :: tmp
    if (y <= (-1.4d+161)) then
        tmp = 2.0d0
    else if ((y <= (-1.22d+87)) .or. (.not. (y <= (-4.5d+60))) .and. (y <= 1d-30)) then
        tmp = 1.0d0 + (z * ((-4.0d0) / y))
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+161) {
		tmp = 2.0;
	} else if ((y <= -1.22e+87) || (!(y <= -4.5e+60) && (y <= 1e-30))) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+161:
		tmp = 2.0
	elif (y <= -1.22e+87) or (not (y <= -4.5e+60) and (y <= 1e-30)):
		tmp = 1.0 + (z * (-4.0 / y))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+161)
		tmp = 2.0;
	elseif ((y <= -1.22e+87) || (!(y <= -4.5e+60) && (y <= 1e-30)))
		tmp = Float64(1.0 + Float64(z * Float64(-4.0 / y)));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e+161)
		tmp = 2.0;
	elseif ((y <= -1.22e+87) || (~((y <= -4.5e+60)) && (y <= 1e-30)))
		tmp = 1.0 + (z * (-4.0 / y));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+161], 2.0, If[Or[LessEqual[y, -1.22e+87], And[N[Not[LessEqual[y, -4.5e+60]], $MachinePrecision], LessEqual[y, 1e-30]]], N[(1.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4000000000000001e161 or -1.2200000000000001e87 < y < -4.50000000000000013e60 or 1e-30 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.8%

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

   if -1.4000000000000001e161 < y < -1.2200000000000001e87 or -4.50000000000000013e60 < y < 1e-30

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 51.7%

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

      1. associate-*r/ 51.7%

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

      2. metadata-eval 51.7%

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

      3. associate-*r* 51.7%

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

      4. neg-mul-1 51.7%

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

      5. *-commutative 51.7%

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

      6. associate-*r/ 51.6%

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

      7. distribute-lft-neg-out 51.6%

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

      8. distribute-rgt-neg-in 51.6%

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

      9. distribute-neg-frac 51.6%

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

      10. metadata-eval 51.6%

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

   5. Simplified 51.6%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{-4}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+161}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{+87} \lor \neg \left(y \leq -4.5 \cdot 10^{+60}\right) \land y \leq 10^{-30}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+234} \lor \neg \left(x \leq 4.7 \cdot 10^{+87}\right):\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e+234) (not (<= x 4.7e+87)))
   (+ 1.0 (/ (* x 4.0) y))
   (+ 2.0 (* -4.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+234) || !(x <= 4.7e+87)) {
		tmp = 1.0 + ((x * 4.0) / y);
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	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) :: tmp
    if ((x <= (-1.02d+234)) .or. (.not. (x <= 4.7d+87))) then
        tmp = 1.0d0 + ((x * 4.0d0) / y)
    else
        tmp = 2.0d0 + ((-4.0d0) * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+234) || !(x <= 4.7e+87)) {
		tmp = 1.0 + ((x * 4.0) / y);
	} else {
		tmp = 2.0 + (-4.0 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e+234) or not (x <= 4.7e+87):
		tmp = 1.0 + ((x * 4.0) / y)
	else:
		tmp = 2.0 + (-4.0 * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e+234) || !(x <= 4.7e+87))
		tmp = Float64(1.0 + Float64(Float64(x * 4.0) / y));
	else
		tmp = Float64(2.0 + Float64(-4.0 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.02e+234) || ~((x <= 4.7e+87)))
		tmp = 1.0 + ((x * 4.0) / y);
	else
		tmp = 2.0 + (-4.0 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e+234], N[Not[LessEqual[x, 4.7e+87]], $MachinePrecision]], N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02000000000000002e234 or 4.7000000000000004e87 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l/ 82.9%

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

   5. Simplified 82.9%

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

   if -1.02000000000000002e234 < x < 4.7000000000000004e87

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} + 1} \]

      2. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} + 1 \]

      3. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) + 1 \]

      4. associate--l+ 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} + 1 \]

      5. +-commutative 99.7%

         \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} + 1 \]

      6. distribute-lft-in 99.8%

         \[\leadsto \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} + 1 \]

      7. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + \left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + 1\right)} \]

      8. associate-*l/ 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\color{blue}{\frac{4 \cdot \left(y \cdot 0.25\right)}{y}} + 1\right) \]

      9. *-commutative 99.8%

         \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{\left(y \cdot 0.25\right) \cdot 4}}{y} + 1\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{y \cdot \left(0.25 \cdot 4\right)}}{y} + 1\right) \]

      11. metadata-eval 99.8%

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

      12. *-rgt-identity 99.8%

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

      13. *-inverses 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + 2} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.8%

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

      2. div-inv 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around 0 80.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{z}{y}} + 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+234} \lor \neg \left(x \leq 4.7 \cdot 10^{+87}\right):\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1060000000000 \lor \neg \left(x \leq 1.35 \cdot 10^{-22}\right):\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1060000000000.0) (not (<= x 1.35e-22)))
   (+ 1.0 (/ (* x 4.0) y))
   (+ (* -4.0 (/ z y)) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1060000000000.0) || !(x <= 1.35e-22)) {
		tmp = 1.0 + ((x * 4.0) / y);
	} else {
		tmp = (-4.0 * (z / y)) + 1.0;
	}
	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) :: tmp
    if ((x <= (-1060000000000.0d0)) .or. (.not. (x <= 1.35d-22))) then
        tmp = 1.0d0 + ((x * 4.0d0) / y)
    else
        tmp = ((-4.0d0) * (z / y)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1060000000000.0) || !(x <= 1.35e-22)) {
		tmp = 1.0 + ((x * 4.0) / y);
	} else {
		tmp = (-4.0 * (z / y)) + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1060000000000.0) or not (x <= 1.35e-22):
		tmp = 1.0 + ((x * 4.0) / y)
	else:
		tmp = (-4.0 * (z / y)) + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1060000000000.0) || !(x <= 1.35e-22))
		tmp = Float64(1.0 + Float64(Float64(x * 4.0) / y));
	else
		tmp = Float64(Float64(-4.0 * Float64(z / y)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1060000000000.0) || ~((x <= 1.35e-22)))
		tmp = 1.0 + ((x * 4.0) / y);
	else
		tmp = (-4.0 * (z / y)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1060000000000.0], N[Not[LessEqual[x, 1.35e-22]], $MachinePrecision]], N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.06e12 or 1.3500000000000001e-22 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l/ 62.2%

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

   5. Simplified 62.2%

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

   if -1.06e12 < x < 1.3500000000000001e-22

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 59.4%

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

      1. *-commutative 59.4%

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

   5. Simplified 59.4%

      \[\leadsto 1 + \color{blue}{\frac{z}{y} \cdot -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1060000000000 \lor \neg \left(x \leq 1.35 \cdot 10^{-22}\right):\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{z}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 2 + \left(x - z\right) \cdot \frac{4}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 2.0 (* (- x z) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	return 2.0 + ((x - z) * (4.0 / 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 = 2.0d0 + ((x - z) * (4.0d0 / y))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 + ((x - z) * (4.0 / y));
}

Python

def code(x, y, z):
	return 2.0 + ((x - z) * (4.0 / y))

Julia

function code(x, y, z)
	return Float64(2.0 + Float64(Float64(x - z) * Float64(4.0 / y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 + ((x - z) * (4.0 / y));
end

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} + 1} \]

   2. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} + 1 \]

   3. +-commutative 99.7%

      \[\leadsto \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) + 1 \]

   4. associate--l+ 99.7%

      \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} + 1 \]

   5. +-commutative 99.7%

      \[\leadsto \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} + 1 \]

   6. distribute-lft-in 99.8%

      \[\leadsto \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} + 1 \]

   7. associate-+l+ 99.8%

      \[\leadsto \color{blue}{\frac{4}{y} \cdot \left(x - z\right) + \left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + 1\right)} \]

   8. associate-*l/ 99.8%

      \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\color{blue}{\frac{4 \cdot \left(y \cdot 0.25\right)}{y}} + 1\right) \]

   9. *-commutative 99.8%

      \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{\left(y \cdot 0.25\right) \cdot 4}}{y} + 1\right) \]

   10. associate-*l* 99.8%

       \[\leadsto \frac{4}{y} \cdot \left(x - z\right) + \left(\frac{\color{blue}{y \cdot \left(0.25 \cdot 4\right)}}{y} + 1\right) \]

   11. metadata-eval 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. *-inverses 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 34.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 2.0)

C

double code(double x, double y, double z) {
	return 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 2.0;
}

Python

def code(x, y, z):
	return 2.0

Julia

function code(x, y, z)
	return 2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0;
end

Wolfram

code[x_, y_, z_] := 2.0

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 31.4%

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

Alternative 9: 8.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 42.7%

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

   1. associate-*r/ 42.7%

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

   2. metadata-eval 42.7%

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

   3. associate-*r* 42.7%

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

   4. neg-mul-1 42.7%

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

   5. *-commutative 42.7%

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

   6. associate-*r/ 42.6%

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

   7. distribute-lft-neg-out 42.6%

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

   8. distribute-rgt-neg-in 42.6%

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

   9. distribute-neg-frac 42.6%

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

   10. metadata-eval 42.6%

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

5. Simplified 42.6%

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

6. Taylor expanded in z around 0 8.2%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))