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

Percentage Accurate: 99.7% → 100.0%

Time: 6.5s

Alternatives: 9

Speedup: 1.4×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+178}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+59} \lor \neg \left(y \leq 2.9 \cdot 10^{+93}\right) \land y \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.3e+178)
   4.0
   (if (or (<= y 9.2e+59) (and (not (<= y 2.9e+93)) (<= y 1.9e+140)))
     (* 4.0 (/ (- x z) y))
     4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.3e+178) {
		tmp = 4.0;
	} else if ((y <= 9.2e+59) || (!(y <= 2.9e+93) && (y <= 1.9e+140))) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.3d+178)) then
        tmp = 4.0d0
    else if ((y <= 9.2d+59) .or. (.not. (y <= 2.9d+93)) .and. (y <= 1.9d+140)) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.3e+178) {
		tmp = 4.0;
	} else if ((y <= 9.2e+59) || (!(y <= 2.9e+93) && (y <= 1.9e+140))) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.3e+178:
		tmp = 4.0
	elif (y <= 9.2e+59) or (not (y <= 2.9e+93) and (y <= 1.9e+140)):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.3e+178)
		tmp = 4.0;
	elseif ((y <= 9.2e+59) || (!(y <= 2.9e+93) && (y <= 1.9e+140)))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.3e+178)
		tmp = 4.0;
	elseif ((y <= 9.2e+59) || (~((y <= 2.9e+93)) && (y <= 1.9e+140)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.3e+178], 4.0, If[Or[LessEqual[y, 9.2e+59], And[N[Not[LessEqual[y, 2.9e+93]], $MachinePrecision], LessEqual[y, 1.9e+140]]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.29999999999999988e178 or 9.20000000000000032e59 < y < 2.8999999999999998e93 or 1.9e140 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 84.9%

      \[\leadsto \color{blue}{4} \]

   if -5.29999999999999988e178 < y < 9.20000000000000032e59 or 2.8999999999999998e93 < y < 1.9e140

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 47.5%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in y around 0 80.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+178}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+59} \lor \neg \left(y \leq 2.9 \cdot 10^{+93}\right) \land y \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 3: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2600:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -9.5e-29)
     t_0
     (if (<= x 2600.0) 4.0 (if (<= x 1.9e+66) (* (/ z y) -4.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -9.5e-29) {
		tmp = t_0;
	} else if (x <= 2600.0) {
		tmp = 4.0;
	} else if (x <= 1.9e+66) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    if (x <= (-9.5d-29)) then
        tmp = t_0
    else if (x <= 2600.0d0) then
        tmp = 4.0d0
    else if (x <= 1.9d+66) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -9.5e-29) {
		tmp = t_0;
	} else if (x <= 2600.0) {
		tmp = 4.0;
	} else if (x <= 1.9e+66) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	tmp = 0
	if x <= -9.5e-29:
		tmp = t_0
	elif x <= 2600.0:
		tmp = 4.0
	elif x <= 1.9e+66:
		tmp = (z / y) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -9.5e-29)
		tmp = t_0;
	elseif (x <= 2600.0)
		tmp = 4.0;
	elseif (x <= 1.9e+66)
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -9.5e-29)
		tmp = t_0;
	elseif (x <= 2600.0)
		tmp = 4.0;
	elseif (x <= 1.9e+66)
		tmp = (z / y) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e-29], t$95$0, If[LessEqual[x, 2600.0], 4.0, If[LessEqual[x, 1.9e+66], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.50000000000000023e-29 or 1.9000000000000001e66 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 47.4%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in z around inf 75.7%

      \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

   5. Taylor expanded in x around inf 66.5%

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

   if -9.50000000000000023e-29 < x < 2600

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 56.2%

      \[\leadsto \color{blue}{4} \]

   if 2600 < x < 1.9000000000000001e66

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 46.6%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

   5. Taylor expanded in z around inf 68.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2600:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 420:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -3.1e-29)
     t_0
     (if (<= x 420.0) 4.0 (if (<= x 7.4e+68) (* z (/ -4.0 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -3.1e-29) {
		tmp = t_0;
	} else if (x <= 420.0) {
		tmp = 4.0;
	} else if (x <= 7.4e+68) {
		tmp = z * (-4.0 / y);
	} 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) :: tmp
    t_0 = 4.0d0 * (x / y)
    if (x <= (-3.1d-29)) then
        tmp = t_0
    else if (x <= 420.0d0) then
        tmp = 4.0d0
    else if (x <= 7.4d+68) then
        tmp = z * ((-4.0d0) / y)
    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 * (x / y);
	double tmp;
	if (x <= -3.1e-29) {
		tmp = t_0;
	} else if (x <= 420.0) {
		tmp = 4.0;
	} else if (x <= 7.4e+68) {
		tmp = z * (-4.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	tmp = 0
	if x <= -3.1e-29:
		tmp = t_0
	elif x <= 420.0:
		tmp = 4.0
	elif x <= 7.4e+68:
		tmp = z * (-4.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -3.1e-29)
		tmp = t_0;
	elseif (x <= 420.0)
		tmp = 4.0;
	elseif (x <= 7.4e+68)
		tmp = Float64(z * Float64(-4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -3.1e-29)
		tmp = t_0;
	elseif (x <= 420.0)
		tmp = 4.0;
	elseif (x <= 7.4e+68)
		tmp = z * (-4.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e-29], t$95$0, If[LessEqual[x, 420.0], 4.0, If[LessEqual[x, 7.4e+68], N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000026e-29 or 7.39999999999999996e68 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 47.4%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in z around inf 75.7%

      \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

   5. Taylor expanded in x around inf 66.5%

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

   if -3.10000000000000026e-29 < x < 420

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 56.2%

      \[\leadsto \color{blue}{4} \]

   if 420 < x < 7.39999999999999996e68

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 46.6%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

   5. Taylor expanded in z around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. associate-/l* 68.8%

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

      3. associate-/r/ 68.9%

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

   7. Simplified 68.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 420:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 85.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-30} \lor \neg \left(x \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e-30) (not (<= x 2e+64)))
   (* 4.0 (/ (- x z) y))
   (+ 4.0 (* (/ z y) -4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.5e-30) || !(x <= 2e+64)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + ((z / y) * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-6.5d-30)) .or. (.not. (x <= 2d+64))) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0 + ((z / y) * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.5e-30) || !(x <= 2e+64)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + ((z / y) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.5e-30) or not (x <= 2e+64):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0 + ((z / y) * -4.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.5e-30) || !(x <= 2e+64))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = Float64(4.0 + Float64(Float64(z / y) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.5e-30) || ~((x <= 2e+64)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0 + ((z / y) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.5e-30], N[Not[LessEqual[x, 2e+64]], $MachinePrecision]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000005e-30 or 2.00000000000000004e64 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 47.9%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in y around 0 81.9%

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

   if -6.5000000000000005e-30 < x < 2.00000000000000004e64

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.75 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.75\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      9. associate-*r* 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(\frac{4}{y} \cdot y\right) \cdot 0.75}\right) \]

      10. associate-*l/ 99.9%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4 \cdot y}{y}} \cdot 0.75\right) \]

      11. associate-/l* 99.9%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4}{\frac{y}{y}}} \cdot 0.75\right) \]

      12. *-inverses 99.9%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{\color{blue}{1}} \cdot 0.75\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{4} \cdot 0.75\right) \]

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 92.4%

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

      1. *-commutative 92.4%

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

   6. Simplified 92.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-30} \lor \neg \left(x \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 6: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+44)
   (+ 4.0 (* (/ z y) -4.0))
   (if (<= z 1.9e+78) (+ 4.0 (/ (* 4.0 x) y)) (* 4.0 (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+44) {
		tmp = 4.0 + ((z / y) * -4.0);
	} else if (z <= 1.9e+78) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.6d+44)) then
        tmp = 4.0d0 + ((z / y) * (-4.0d0))
    else if (z <= 1.9d+78) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 * ((x - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+44) {
		tmp = 4.0 + ((z / y) * -4.0);
	} else if (z <= 1.9e+78) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.6e+44:
		tmp = 4.0 + ((z / y) * -4.0)
	elif z <= 1.9e+78:
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 * ((x - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+44)
		tmp = Float64(4.0 + Float64(Float64(z / y) * -4.0));
	elseif (z <= 1.9e+78)
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.6e+44)
		tmp = 4.0 + ((z / y) * -4.0);
	elseif (z <= 1.9e+78)
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 * ((x - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+44], N[(4.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e+78], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5999999999999999e44

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.75 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.75\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.7%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.7%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      8. fma-def 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      9. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(\frac{4}{y} \cdot y\right) \cdot 0.75}\right) \]

      10. associate-*l/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4 \cdot y}{y}} \cdot 0.75\right) \]

      11. associate-/l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4}{\frac{y}{y}}} \cdot 0.75\right) \]

      12. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{\color{blue}{1}} \cdot 0.75\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{4} \cdot 0.75\right) \]

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 88.0%

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

      1. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   if -2.5999999999999999e44 < z < 1.9e78

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      3. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.75 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.75\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.7%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.7%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      8. fma-def 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.75\right)\right)} \]

      9. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(\frac{4}{y} \cdot y\right) \cdot 0.75}\right) \]

      10. associate-*l/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4 \cdot y}{y}} \cdot 0.75\right) \]

      11. associate-/l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\frac{4}{\frac{y}{y}}} \cdot 0.75\right) \]

      12. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{\color{blue}{1}} \cdot 0.75\right) \]

      13. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{4} \cdot 0.75\right) \]

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 92.3%

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

      1. associate-*r/ 92.3%

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

   6. Simplified 92.3%

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

   if 1.9e78 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 30.6%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in y around 0 90.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \]

Alternative 7: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+115} \lor \neg \left(z \leq 3 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.95e+115) (not (<= z 3e+99))) (* (/ z y) -4.0) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+115) || !(z <= 3e+99)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.95d+115)) .or. (.not. (z <= 3d+99))) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+115) || !(z <= 3e+99)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.95e+115) or not (z <= 3e+99):
		tmp = (z / y) * -4.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.95e+115) || !(z <= 3e+99))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.95e+115) || ~((z <= 3e+99)))
		tmp = (z / y) * -4.0;
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.95e+115], N[Not[LessEqual[z, 3e+99]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if z < -1.95000000000000003e115 or 3.00000000000000014e99 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. flip-+ 30.5%

         \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

   3. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

   4. Taylor expanded in z around inf 84.4%

      \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

   5. Taylor expanded in z around inf 79.3%

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

   if -1.95000000000000003e115 < z < 3.00000000000000014e99

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 46.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+115} \lor \neg \left(z \leq 3 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 8: 2.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -3.0)

C

double code(double x, double y, double z) {
	return -3.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -3.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -3.0;
}

Python

def code(x, y, z):
	return -3.0

Julia

function code(x, y, z)
	return -3.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -3.0;
end

Wolfram

code[x_, y_, z_] := -3.0

Derivation

1. Initial program 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

   2. flip-+ 60.3%

      \[\leadsto \color{blue}{\frac{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \cdot \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1 \cdot 1}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} - 1}} \]

3. Applied egg-rr 59.8%

   \[\leadsto \color{blue}{\frac{\frac{16}{{\left(\frac{y}{\mathsf{fma}\left(y, 0.75, x\right) - z}\right)}^{2}} - 1}{\left(\mathsf{fma}\left(y, 0.75, x\right) - z\right) \cdot \frac{4}{y} - 1}} \]

4. Taylor expanded in z around inf 87.7%

   \[\leadsto \color{blue}{\left(8 \cdot \frac{0.75 \cdot y + x}{y} + \left(1 + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]

5. Taylor expanded in y around 0 52.4%

   \[\leadsto \color{blue}{\frac{-4 \cdot z + 8 \cdot x}{y}} - 4 \cdot \left(0.75 + \frac{x}{y}\right) \]

6. Taylor expanded in y around inf 2.5%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.5%

   \[\leadsto -3 \]

Alternative 9: 33.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 35.7%

   \[\leadsto \color{blue}{4} \]

3. Final simplification 35.7%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))