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

Percentage Accurate: 99.9% → 100.0%

Time: 7.8s

Alternatives: 9

Speedup: 1.4×

• 20.8% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   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. Final simplification 100.0%

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

Alternative 2: 53.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+84}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-261}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-201}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2200:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (* (/ z y) -4.0)))
   (if (<= y -8.5e+84)
     2.0
     (if (<= y -6.4e+18)
       t_1
       (if (<= y -8.6e-8)
         2.0
         (if (<= y -4.3e-193)
           t_0
           (if (<= y -1.6e-261)
             t_1
             (if (<= y -8.5e-280)
               t_0
               (if (<= y 2.8e-201)
                 t_1
                 (if (<= y 5.8e-146)
                   t_0
                   (if (<= y 2200.0) t_1 (if (<= y 1.1e+62) t_0 2.0))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (z / y) * -4.0;
	double tmp;
	if (y <= -8.5e+84) {
		tmp = 2.0;
	} else if (y <= -6.4e+18) {
		tmp = t_1;
	} else if (y <= -8.6e-8) {
		tmp = 2.0;
	} else if (y <= -4.3e-193) {
		tmp = t_0;
	} else if (y <= -1.6e-261) {
		tmp = t_1;
	} else if (y <= -8.5e-280) {
		tmp = t_0;
	} else if (y <= 2.8e-201) {
		tmp = t_1;
	} else if (y <= 5.8e-146) {
		tmp = t_0;
	} else if (y <= 2200.0) {
		tmp = t_1;
	} else if (y <= 1.1e+62) {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    t_1 = (z / y) * (-4.0d0)
    if (y <= (-8.5d+84)) then
        tmp = 2.0d0
    else if (y <= (-6.4d+18)) then
        tmp = t_1
    else if (y <= (-8.6d-8)) then
        tmp = 2.0d0
    else if (y <= (-4.3d-193)) then
        tmp = t_0
    else if (y <= (-1.6d-261)) then
        tmp = t_1
    else if (y <= (-8.5d-280)) then
        tmp = t_0
    else if (y <= 2.8d-201) then
        tmp = t_1
    else if (y <= 5.8d-146) then
        tmp = t_0
    else if (y <= 2200.0d0) then
        tmp = t_1
    else if (y <= 1.1d+62) then
        tmp = t_0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (z / y) * -4.0;
	double tmp;
	if (y <= -8.5e+84) {
		tmp = 2.0;
	} else if (y <= -6.4e+18) {
		tmp = t_1;
	} else if (y <= -8.6e-8) {
		tmp = 2.0;
	} else if (y <= -4.3e-193) {
		tmp = t_0;
	} else if (y <= -1.6e-261) {
		tmp = t_1;
	} else if (y <= -8.5e-280) {
		tmp = t_0;
	} else if (y <= 2.8e-201) {
		tmp = t_1;
	} else if (y <= 5.8e-146) {
		tmp = t_0;
	} else if (y <= 2200.0) {
		tmp = t_1;
	} else if (y <= 1.1e+62) {
		tmp = t_0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = (z / y) * -4.0
	tmp = 0
	if y <= -8.5e+84:
		tmp = 2.0
	elif y <= -6.4e+18:
		tmp = t_1
	elif y <= -8.6e-8:
		tmp = 2.0
	elif y <= -4.3e-193:
		tmp = t_0
	elif y <= -1.6e-261:
		tmp = t_1
	elif y <= -8.5e-280:
		tmp = t_0
	elif y <= 2.8e-201:
		tmp = t_1
	elif y <= 5.8e-146:
		tmp = t_0
	elif y <= 2200.0:
		tmp = t_1
	elif y <= 1.1e+62:
		tmp = t_0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(Float64(z / y) * -4.0)
	tmp = 0.0
	if (y <= -8.5e+84)
		tmp = 2.0;
	elseif (y <= -6.4e+18)
		tmp = t_1;
	elseif (y <= -8.6e-8)
		tmp = 2.0;
	elseif (y <= -4.3e-193)
		tmp = t_0;
	elseif (y <= -1.6e-261)
		tmp = t_1;
	elseif (y <= -8.5e-280)
		tmp = t_0;
	elseif (y <= 2.8e-201)
		tmp = t_1;
	elseif (y <= 5.8e-146)
		tmp = t_0;
	elseif (y <= 2200.0)
		tmp = t_1;
	elseif (y <= 1.1e+62)
		tmp = t_0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = (z / y) * -4.0;
	tmp = 0.0;
	if (y <= -8.5e+84)
		tmp = 2.0;
	elseif (y <= -6.4e+18)
		tmp = t_1;
	elseif (y <= -8.6e-8)
		tmp = 2.0;
	elseif (y <= -4.3e-193)
		tmp = t_0;
	elseif (y <= -1.6e-261)
		tmp = t_1;
	elseif (y <= -8.5e-280)
		tmp = t_0;
	elseif (y <= 2.8e-201)
		tmp = t_1;
	elseif (y <= 5.8e-146)
		tmp = t_0;
	elseif (y <= 2200.0)
		tmp = t_1;
	elseif (y <= 1.1e+62)
		tmp = t_0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[y, -8.5e+84], 2.0, If[LessEqual[y, -6.4e+18], t$95$1, If[LessEqual[y, -8.6e-8], 2.0, If[LessEqual[y, -4.3e-193], t$95$0, If[LessEqual[y, -1.6e-261], t$95$1, If[LessEqual[y, -8.5e-280], t$95$0, If[LessEqual[y, 2.8e-201], t$95$1, If[LessEqual[y, 5.8e-146], t$95$0, If[LessEqual[y, 2200.0], t$95$1, If[LessEqual[y, 1.1e+62], t$95$0, 2.0]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000008e84 or -6.4e18 < y < -8.6000000000000002e-8 or 1.10000000000000007e62 < 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 78.2%

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

   if -8.5000000000000008e84 < y < -6.4e18 or -4.3000000000000002e-193 < y < -1.60000000000000002e-261 or -8.50000000000000037e-280 < y < 2.7999999999999999e-201 or 5.80000000000000022e-146 < y < 2200

   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 99.7%

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

      7. associate-+l+ 99.7%

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

      8. associate-*l/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-rgt-identity 99.7%

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

      13. *-inverses 99.7%

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

      14. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      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 z around inf 70.3%

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

   if -8.6000000000000002e-8 < y < -4.3000000000000002e-193 or -1.60000000000000002e-261 < y < -8.50000000000000037e-280 or 2.7999999999999999e-201 < y < 5.80000000000000022e-146 or 2200 < y < 1.10000000000000007e62

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      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 99.7%

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

      7. associate-+l+ 99.7%

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

      8. associate-*l/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-rgt-identity 99.7%

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

      13. *-inverses 99.7%

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

      14. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      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 68.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+84}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-193}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-261}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-280}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2200:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := 1 + z \cdot \frac{-4}{y}\\ t_2 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.46 \cdot 10^{-257}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-281}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-200}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y)))
        (t_1 (+ 1.0 (* z (/ -4.0 y))))
        (t_2 (* (/ z y) -4.0)))
   (if (<= y -5.8e+84)
     2.0
     (if (<= y -9e-13)
       t_1
       (if (<= y -1.35e-191)
         t_0
         (if (<= y -1.46e-257)
           t_2
           (if (<= y -3.8e-281)
             t_0
             (if (<= y 1.85e-200)
               t_2
               (if (<= y 3e-147) t_0 (if (<= y 3.2e+62) t_1 2.0))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = 1.0 + (z * (-4.0 / y));
	double t_2 = (z / y) * -4.0;
	double tmp;
	if (y <= -5.8e+84) {
		tmp = 2.0;
	} else if (y <= -9e-13) {
		tmp = t_1;
	} else if (y <= -1.35e-191) {
		tmp = t_0;
	} else if (y <= -1.46e-257) {
		tmp = t_2;
	} else if (y <= -3.8e-281) {
		tmp = t_0;
	} else if (y <= 1.85e-200) {
		tmp = t_2;
	} else if (y <= 3e-147) {
		tmp = t_0;
	} else if (y <= 3.2e+62) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    t_1 = 1.0d0 + (z * ((-4.0d0) / y))
    t_2 = (z / y) * (-4.0d0)
    if (y <= (-5.8d+84)) then
        tmp = 2.0d0
    else if (y <= (-9d-13)) then
        tmp = t_1
    else if (y <= (-1.35d-191)) then
        tmp = t_0
    else if (y <= (-1.46d-257)) then
        tmp = t_2
    else if (y <= (-3.8d-281)) then
        tmp = t_0
    else if (y <= 1.85d-200) then
        tmp = t_2
    else if (y <= 3d-147) then
        tmp = t_0
    else if (y <= 3.2d+62) then
        tmp = t_1
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = 1.0 + (z * (-4.0 / y));
	double t_2 = (z / y) * -4.0;
	double tmp;
	if (y <= -5.8e+84) {
		tmp = 2.0;
	} else if (y <= -9e-13) {
		tmp = t_1;
	} else if (y <= -1.35e-191) {
		tmp = t_0;
	} else if (y <= -1.46e-257) {
		tmp = t_2;
	} else if (y <= -3.8e-281) {
		tmp = t_0;
	} else if (y <= 1.85e-200) {
		tmp = t_2;
	} else if (y <= 3e-147) {
		tmp = t_0;
	} else if (y <= 3.2e+62) {
		tmp = t_1;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = 1.0 + (z * (-4.0 / y))
	t_2 = (z / y) * -4.0
	tmp = 0
	if y <= -5.8e+84:
		tmp = 2.0
	elif y <= -9e-13:
		tmp = t_1
	elif y <= -1.35e-191:
		tmp = t_0
	elif y <= -1.46e-257:
		tmp = t_2
	elif y <= -3.8e-281:
		tmp = t_0
	elif y <= 1.85e-200:
		tmp = t_2
	elif y <= 3e-147:
		tmp = t_0
	elif y <= 3.2e+62:
		tmp = t_1
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(1.0 + Float64(z * Float64(-4.0 / y)))
	t_2 = Float64(Float64(z / y) * -4.0)
	tmp = 0.0
	if (y <= -5.8e+84)
		tmp = 2.0;
	elseif (y <= -9e-13)
		tmp = t_1;
	elseif (y <= -1.35e-191)
		tmp = t_0;
	elseif (y <= -1.46e-257)
		tmp = t_2;
	elseif (y <= -3.8e-281)
		tmp = t_0;
	elseif (y <= 1.85e-200)
		tmp = t_2;
	elseif (y <= 3e-147)
		tmp = t_0;
	elseif (y <= 3.2e+62)
		tmp = t_1;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = 1.0 + (z * (-4.0 / y));
	t_2 = (z / y) * -4.0;
	tmp = 0.0;
	if (y <= -5.8e+84)
		tmp = 2.0;
	elseif (y <= -9e-13)
		tmp = t_1;
	elseif (y <= -1.35e-191)
		tmp = t_0;
	elseif (y <= -1.46e-257)
		tmp = t_2;
	elseif (y <= -3.8e-281)
		tmp = t_0;
	elseif (y <= 1.85e-200)
		tmp = t_2;
	elseif (y <= 3e-147)
		tmp = t_0;
	elseif (y <= 3.2e+62)
		tmp = t_1;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[y, -5.8e+84], 2.0, If[LessEqual[y, -9e-13], t$95$1, If[LessEqual[y, -1.35e-191], t$95$0, If[LessEqual[y, -1.46e-257], t$95$2, If[LessEqual[y, -3.8e-281], t$95$0, If[LessEqual[y, 1.85e-200], t$95$2, If[LessEqual[y, 3e-147], t$95$0, If[LessEqual[y, 3.2e+62], t$95$1, 2.0]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.79999999999999977e84 or 3.19999999999999984e62 < 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 79.7%

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

   if -5.79999999999999977e84 < y < -9e-13 or 3.0000000000000002e-147 < y < 3.19999999999999984e62

   1. Initial program 98.7%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   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. associate-*r/ 59.4%

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

      2. metadata-eval 59.4%

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

      3. associate-*r* 59.4%

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

      4. neg-mul-1 59.4%

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

      5. associate-*l/ 59.2%

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

      6. metadata-eval 59.2%

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

      7. associate-*r/ 59.2%

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

      8. distribute-rgt-neg-in 59.2%

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

      9. *-commutative 59.2%

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

      10. distribute-rgt-neg-in 59.2%

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

      11. associate-*r/ 59.2%

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

      12. metadata-eval 59.2%

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

      13. distribute-neg-frac 59.2%

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

      14. metadata-eval 59.2%

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

   5. Simplified 59.2%

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

   if -9e-13 < y < -1.34999999999999999e-191 or -1.4600000000000001e-257 < y < -3.79999999999999976e-281 or 1.85000000000000005e-200 < y < 3.0000000000000002e-147

   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 99.6%

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

      7. associate-+l+ 99.7%

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

      8. associate-*l/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-rgt-identity 99.7%

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

      13. *-inverses 99.7%

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

      14. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      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 72.0%

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

   if -1.34999999999999999e-191 < y < -1.4600000000000001e-257 or -3.79999999999999976e-281 < y < 1.85000000000000005e-200

   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/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      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 z around inf 71.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-13}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-191}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.46 \cdot 10^{-257}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-281}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-200}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-147}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+40} \lor \neg \left(z \leq 10^{-10} \lor \neg \left(z \leq 8 \cdot 10^{+14}\right) \land z \leq 1.65 \cdot 10^{+158}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e+40)
         (not (or (<= z 1e-10) (and (not (<= z 8e+14)) (<= z 1.65e+158)))))
   (+ 1.0 (* z (/ -4.0 y)))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+40) || !((z <= 1e-10) || (!(z <= 8e+14) && (z <= 1.65e+158)))) {
		tmp = 1.0 + (z * (-4.0 / 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 <= (-1.15d+40)) .or. (.not. (z <= 1d-10) .or. (.not. (z <= 8d+14)) .and. (z <= 1.65d+158))) then
        tmp = 1.0d0 + (z * ((-4.0d0) / 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 <= -1.15e+40) || !((z <= 1e-10) || (!(z <= 8e+14) && (z <= 1.65e+158)))) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e+40) or not ((z <= 1e-10) or (not (z <= 8e+14) and (z <= 1.65e+158))):
		tmp = 1.0 + (z * (-4.0 / y))
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e+40) || !((z <= 1e-10) || (!(z <= 8e+14) && (z <= 1.65e+158))))
		tmp = Float64(1.0 + Float64(z * Float64(-4.0 / 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 <= -1.15e+40) || ~(((z <= 1e-10) || (~((z <= 8e+14)) && (z <= 1.65e+158)))))
		tmp = 1.0 + (z * (-4.0 / y));
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e+40], N[Not[Or[LessEqual[z, 1e-10], And[N[Not[LessEqual[z, 8e+14]], $MachinePrecision], LessEqual[z, 1.65e+158]]]], $MachinePrecision]], N[(1.0 + N[(z * N[(-4.0 / 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 < -1.14999999999999997e40 or 1.00000000000000004e-10 < z < 8e14 or 1.65000000000000009e158 < z

   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 81.4%

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

      1. associate-*r/ 81.4%

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

      2. metadata-eval 81.4%

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

      3. associate-*r* 81.4%

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

      4. neg-mul-1 81.4%

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

      5. associate-*l/ 81.2%

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

      6. metadata-eval 81.2%

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

      7. associate-*r/ 81.2%

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

      8. distribute-rgt-neg-in 81.2%

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

      9. *-commutative 81.2%

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

      10. distribute-rgt-neg-in 81.2%

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

      11. associate-*r/ 81.2%

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

      12. metadata-eval 81.2%

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

      13. distribute-neg-frac 81.2%

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

      14. metadata-eval 81.2%

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

   5. Simplified 81.2%

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

   if -1.14999999999999997e40 < z < 1.00000000000000004e-10 or 8e14 < z < 1.65000000000000009e158

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      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. Taylor expanded in z around 0 89.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+40} \lor \neg \left(z \leq 10^{-10} \lor \neg \left(z \leq 8 \cdot 10^{+14}\right) \land z \leq 1.65 \cdot 10^{+158}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+83}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+20} \lor \neg \left(y \leq -7.5 \cdot 10^{-74}\right) \land y \leq 7.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.4e+83)
   2.0
   (if (or (<= y -3.2e+20) (and (not (<= y -7.5e-74)) (<= y 7.8e+62)))
     (* (/ z y) -4.0)
     2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e+83) {
		tmp = 2.0;
	} else if ((y <= -3.2e+20) || (!(y <= -7.5e-74) && (y <= 7.8e+62))) {
		tmp = (z / y) * -4.0;
	} 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 <= (-4.4d+83)) then
        tmp = 2.0d0
    else if ((y <= (-3.2d+20)) .or. (.not. (y <= (-7.5d-74))) .and. (y <= 7.8d+62)) then
        tmp = (z / y) * (-4.0d0)
    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 <= -4.4e+83) {
		tmp = 2.0;
	} else if ((y <= -3.2e+20) || (!(y <= -7.5e-74) && (y <= 7.8e+62))) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.4e+83:
		tmp = 2.0
	elif (y <= -3.2e+20) or (not (y <= -7.5e-74) and (y <= 7.8e+62)):
		tmp = (z / y) * -4.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.4e+83)
		tmp = 2.0;
	elseif ((y <= -3.2e+20) || (!(y <= -7.5e-74) && (y <= 7.8e+62)))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.4e+83)
		tmp = 2.0;
	elseif ((y <= -3.2e+20) || (~((y <= -7.5e-74)) && (y <= 7.8e+62)))
		tmp = (z / y) * -4.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.4e+83], 2.0, If[Or[LessEqual[y, -3.2e+20], And[N[Not[LessEqual[y, -7.5e-74]], $MachinePrecision], LessEqual[y, 7.8e+62]]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.39999999999999997e83 or -3.2e20 < y < -7.5e-74 or 7.8e62 < 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 74.7%

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

   if -4.39999999999999997e83 < y < -3.2e20 or -7.5e-74 < y < 7.8e62

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      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 99.7%

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

      7. associate-+l+ 99.7%

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

      8. associate-*l/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-rgt-identity 99.7%

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

      13. *-inverses 99.7%

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

      14. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      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 z around inf 52.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+83}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+20} \lor \neg \left(y \leq -7.5 \cdot 10^{-74}\right) \land y \leq 7.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{-17}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.8e+39) (not (<= z 7.8e-17)))
   (+ 2.0 (* (/ z y) -4.0))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+39) || !(z <= 7.8e-17)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} 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 <= (-9.8d+39)) .or. (.not. (z <= 7.8d-17))) then
        tmp = 2.0d0 + ((z / y) * (-4.0d0))
    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 <= -9.8e+39) || !(z <= 7.8e-17)) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.8e+39) or not (z <= 7.8e-17):
		tmp = 2.0 + ((z / y) * -4.0)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.8e+39) || !(z <= 7.8e-17))
		tmp = Float64(2.0 + Float64(Float64(z / y) * -4.0));
	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 <= -9.8e+39) || ~((z <= 7.8e-17)))
		tmp = 2.0 + ((z / y) * -4.0);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.8e+39], N[Not[LessEqual[z, 7.8e-17]], $MachinePrecision]], N[(2.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.79999999999999974e39 or 7.79999999999999979e-17 < 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/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      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. Taylor expanded in x around 0 91.2%

      \[\leadsto \color{blue}{2 + -4 \cdot \frac{z}{y}} \]
   6. Step-by-step derivation

      1. +-commutative 91.2%

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

      2. *-commutative 91.2%

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

   7. Simplified 91.2%

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

   if -9.79999999999999974e39 < z < 7.79999999999999979e-17

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      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. Taylor expanded in z around 0 93.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{-17}\right):\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]
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 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   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: 8.2% 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 99.6%

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

3. Taylor expanded in z around inf 41.3%

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

   1. associate-*r/ 41.3%

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

   2. metadata-eval 41.3%

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

   3. associate-*r* 41.3%

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

   4. neg-mul-1 41.3%

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

   5. associate-*l/ 41.2%

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

   6. metadata-eval 41.2%

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

   7. associate-*r/ 41.2%

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

   8. distribute-rgt-neg-in 41.2%

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

   9. *-commutative 41.2%

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

   10. distribute-rgt-neg-in 41.2%

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

   11. associate-*r/ 41.2%

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

   12. metadata-eval 41.2%

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

   13. distribute-neg-frac 41.2%

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

   14. metadata-eval 41.2%

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

5. Simplified 41.2%

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

6. Taylor expanded in z around 0 8.1%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 8.1%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 9: 34.6% 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 99.6%

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

3. Taylor expanded in y around inf 35.1%

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

4. Final simplification 35.1%

   \[\leadsto 2 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))