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

Percentage Accurate: 99.8% → 100.0%

Time: 4.1s

Alternatives: 7

Speedup: 1.2×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x - y}{z} - 2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

   2. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

   5. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

   6. sub-neg 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

   7. distribute-rgt-neg-in 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

4. Taylor expanded in z around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto 4 \cdot \frac{x - y}{z} - 2 \]

Alternative 2: 52.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-308}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+147}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ y z))) (t_1 (* 4.0 (/ x z))))
   (if (<= y -7.5e+69)
     t_0
     (if (<= y -1.22e-223)
       t_1
       (if (<= y 7e-308)
         -2.0
         (if (<= y 9.6e-140) t_1 (if (<= y 5.1e+147) -2.0 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (y <= -7.5e+69) {
		tmp = t_0;
	} else if (y <= -1.22e-223) {
		tmp = t_1;
	} else if (y <= 7e-308) {
		tmp = -2.0;
	} else if (y <= 9.6e-140) {
		tmp = t_1;
	} else if (y <= 5.1e+147) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-4.0d0) * (y / z)
    t_1 = 4.0d0 * (x / z)
    if (y <= (-7.5d+69)) then
        tmp = t_0
    else if (y <= (-1.22d-223)) then
        tmp = t_1
    else if (y <= 7d-308) then
        tmp = -2.0d0
    else if (y <= 9.6d-140) then
        tmp = t_1
    else if (y <= 5.1d+147) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (y <= -7.5e+69) {
		tmp = t_0;
	} else if (y <= -1.22e-223) {
		tmp = t_1;
	} else if (y <= 7e-308) {
		tmp = -2.0;
	} else if (y <= 9.6e-140) {
		tmp = t_1;
	} else if (y <= 5.1e+147) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (y / z)
	t_1 = 4.0 * (x / z)
	tmp = 0
	if y <= -7.5e+69:
		tmp = t_0
	elif y <= -1.22e-223:
		tmp = t_1
	elif y <= 7e-308:
		tmp = -2.0
	elif y <= 9.6e-140:
		tmp = t_1
	elif y <= 5.1e+147:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(y / z))
	t_1 = Float64(4.0 * Float64(x / z))
	tmp = 0.0
	if (y <= -7.5e+69)
		tmp = t_0;
	elseif (y <= -1.22e-223)
		tmp = t_1;
	elseif (y <= 7e-308)
		tmp = -2.0;
	elseif (y <= 9.6e-140)
		tmp = t_1;
	elseif (y <= 5.1e+147)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (y / z);
	t_1 = 4.0 * (x / z);
	tmp = 0.0;
	if (y <= -7.5e+69)
		tmp = t_0;
	elseif (y <= -1.22e-223)
		tmp = t_1;
	elseif (y <= 7e-308)
		tmp = -2.0;
	elseif (y <= 9.6e-140)
		tmp = t_1;
	elseif (y <= 5.1e+147)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+69], t$95$0, If[LessEqual[y, -1.22e-223], t$95$1, If[LessEqual[y, 7e-308], -2.0, If[LessEqual[y, 9.6e-140], t$95$1, If[LessEqual[y, 5.1e+147], -2.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999939e69 or 5.09999999999999999e147 < y

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in y around inf 75.1%

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

   if -7.49999999999999939e69 < y < -1.21999999999999998e-223 or 7e-308 < y < 9.59999999999999947e-140

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in x around inf 63.6%

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

   if -1.21999999999999998e-223 < y < 7e-308 or 9.59999999999999947e-140 < y < 5.09999999999999999e147

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 53.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+69}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-223}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-308}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-140}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+147}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+29} \lor \neg \left(x \leq 6.1 \cdot 10^{+85}\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.36e+29) (not (<= x 6.1e+85)))
   (* 4.0 (/ (- x y) z))
   (- (* -4.0 (/ y z)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.36e+29) || !(x <= 6.1e+85)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 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 ((x <= (-1.36d+29)) .or. (.not. (x <= 6.1d+85))) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.36e+29) || !(x <= 6.1e+85)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.36e+29) or not (x <= 6.1e+85):
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = (-4.0 * (y / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.36e+29) || !(x <= 6.1e+85))
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.36e+29) || ~((x <= 6.1e+85)))
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = (-4.0 * (y / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.36e+29], N[Not[LessEqual[x, 6.1e+85]], $MachinePrecision]], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.36e29 or 6.09999999999999981e85 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around 0 87.4%

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

   if -1.36e29 < x < 6.09999999999999981e85

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - y}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      11. associate-/r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, -\color{blue}{\frac{z \cdot 0.5}{z} \cdot 4}\right) \]

      12. distribute-lft-neg-in 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\left(-\frac{z \cdot 0.5}{z}\right) \cdot 4}\right) \]

      13. distribute-frac-neg 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-z \cdot 0.5}{z}} \cdot 4\right) \]

      14. *-commutative 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-\color{blue}{0.5 \cdot z}}{z} \cdot 4\right) \]

      15. distribute-lft-neg-in 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{\left(-0.5\right) \cdot z}}{z} \cdot 4\right) \]

      16. associate-/l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-0.5}{\frac{z}{z}}} \cdot 4\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{-0.5}}{\frac{z}{z}} \cdot 4\right) \]

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-0.5} \cdot 4\right) \]

      20. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]

   4. Taylor expanded in x around 0 94.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+29} \lor \neg \left(x \leq 6.1 \cdot 10^{+85}\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \]

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+138}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+74}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e+138) -2.0 (if (<= z 8.5e+74) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+138) {
		tmp = -2.0;
	} else if (z <= 8.5e+74) {
		tmp = 4.0 * ((x - y) / z);
	} 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 (z <= (-2.9d+138)) then
        tmp = -2.0d0
    else if (z <= 8.5d+74) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+138) {
		tmp = -2.0;
	} else if (z <= 8.5e+74) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.9e+138:
		tmp = -2.0
	elif z <= 8.5e+74:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e+138)
		tmp = -2.0;
	elseif (z <= 8.5e+74)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.9e+138)
		tmp = -2.0;
	elseif (z <= 8.5e+74)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e+138], -2.0, If[LessEqual[z, 8.5e+74], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9000000000000001e138 or 8.50000000000000028e74 < z

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 73.1%

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

   if -2.9000000000000001e138 < z < 8.50000000000000028e74

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around 0 87.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+138}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+74}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+27}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e+27)
   (* 4.0 (/ (- x y) z))
   (if (<= x 1.3e+80) (- (* -4.0 (/ y z)) 2.0) (- (* 4.0 (/ x z)) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+27) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 1.3e+80) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / z)) - 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 (x <= (-4.5d+27)) then
        tmp = 4.0d0 * ((x - y) / z)
    else if (x <= 1.3d+80) then
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    else
        tmp = (4.0d0 * (x / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+27) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 1.3e+80) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.5e+27:
		tmp = 4.0 * ((x - y) / z)
	elif x <= 1.3e+80:
		tmp = (-4.0 * (y / z)) - 2.0
	else:
		tmp = (4.0 * (x / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+27)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	elseif (x <= 1.3e+80)
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(x / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.5e+27)
		tmp = 4.0 * ((x - y) / z);
	elseif (x <= 1.3e+80)
		tmp = (-4.0 * (y / z)) - 2.0;
	else
		tmp = (4.0 * (x / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e+27], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+80], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999999e27

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around 0 88.3%

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

   if -4.4999999999999999e27 < x < 1.29999999999999991e80

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - y}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      11. associate-/r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, -\color{blue}{\frac{z \cdot 0.5}{z} \cdot 4}\right) \]

      12. distribute-lft-neg-in 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\left(-\frac{z \cdot 0.5}{z}\right) \cdot 4}\right) \]

      13. distribute-frac-neg 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-z \cdot 0.5}{z}} \cdot 4\right) \]

      14. *-commutative 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-\color{blue}{0.5 \cdot z}}{z} \cdot 4\right) \]

      15. distribute-lft-neg-in 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{\left(-0.5\right) \cdot z}}{z} \cdot 4\right) \]

      16. associate-/l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-0.5}{\frac{z}{z}}} \cdot 4\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{-0.5}}{\frac{z}{z}} \cdot 4\right) \]

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-0.5} \cdot 4\right) \]

      20. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]

   4. Taylor expanded in x around 0 94.6%

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

   if 1.29999999999999991e80 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot 4}}{z} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{\frac{z}{4}}} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x - y}{\frac{z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.9%

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - y}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      11. associate-/r/ 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, -\color{blue}{\frac{z \cdot 0.5}{z} \cdot 4}\right) \]

      12. distribute-lft-neg-in 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\left(-\frac{z \cdot 0.5}{z}\right) \cdot 4}\right) \]

      13. distribute-frac-neg 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-z \cdot 0.5}{z}} \cdot 4\right) \]

      14. *-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-\color{blue}{0.5 \cdot z}}{z} \cdot 4\right) \]

      15. distribute-lft-neg-in 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{\left(-0.5\right) \cdot z}}{z} \cdot 4\right) \]

      16. associate-/l* 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{\frac{-0.5}{\frac{z}{z}}} \cdot 4\right) \]

      17. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{\color{blue}{-0.5}}{\frac{z}{z}} \cdot 4\right) \]

      18. *-inverses 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-0.5} \cdot 4\right) \]

      20. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \color{blue}{-2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]

   4. Taylor expanded in y around 0 92.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+27}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+80}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \]

Alternative 6: 53.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+138}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+74}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+138) -2.0 (if (<= z 5e+74) (* -4.0 (/ y z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+138) {
		tmp = -2.0;
	} else if (z <= 5e+74) {
		tmp = -4.0 * (y / z);
	} 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 (z <= (-1.6d+138)) then
        tmp = -2.0d0
    else if (z <= 5d+74) then
        tmp = (-4.0d0) * (y / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+138) {
		tmp = -2.0;
	} else if (z <= 5e+74) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.6e+138:
		tmp = -2.0
	elif z <= 5e+74:
		tmp = -4.0 * (y / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+138)
		tmp = -2.0;
	elseif (z <= 5e+74)
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.6e+138)
		tmp = -2.0;
	elseif (z <= 5e+74)
		tmp = -4.0 * (y / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+138], -2.0, If[LessEqual[z, 5e+74], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6000000000000001e138 or 4.99999999999999963e74 < z

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 73.1%

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

   if -1.6000000000000001e138 < z < 4.99999999999999963e74

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

      6. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      7. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in y around inf 49.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+138}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+74}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 7: 33.2% accurate, 11.0× 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%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

   2. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

   5. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right)} \]

   6. sub-neg 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

   7. distribute-rgt-neg-in 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

4. Taylor expanded in z around inf 34.2%

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

5. Final simplification 34.2%

   \[\leadsto -2 \]

Developer target: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))