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

Percentage Accurate: 99.8% → 100.0%

Time: 8.9s

Alternatives: 10

Speedup: 1.5×

• 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 100.0%

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

Alternative 2: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+271}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (/ (* 4.0 x) y)))
   (if (<= t_1 -2e+262)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 100000000.0) 4.0 (if (<= t_1 1e+271) t_2 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+271) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (z * (-4.0d0)) / y
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = (4.0d0 * x) / y
    if (t_1 <= (-2d+262)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 100000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 1d+271) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (4.0 * x) / y;
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+271) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -4.0) / y
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = (4.0 * x) / y
	tmp = 0
	if t_1 <= -2e+262:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 100000000.0:
		tmp = 4.0
	elif t_1 <= 1e+271:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+271)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -4.0) / y;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = (4.0 * x) / y;
	tmp = 0.0;
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+271)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+262], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 100000000.0], 4.0, If[LessEqual[t$95$1, 1e+271], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e262 or 9.99999999999999953e270 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 64.9

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

   8. Simplified 64.9%

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

   if -2e262 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999953e270

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 56.4

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

   5. Simplified 56.4%

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

   if -500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 95.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -500:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 10^{+271}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := x \cdot \frac{4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+158}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (* x (/ 4.0 y))))
   (if (<= t_1 -2e+262)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 100000000.0) 4.0 (if (<= t_1 1e+158) t_2 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+158) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (z * (-4.0d0)) / y
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = x * (4.0d0 / y)
    if (t_1 <= (-2d+262)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 100000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 1d+158) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+158) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -4.0) / y
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = x * (4.0 / y)
	tmp = 0
	if t_1 <= -2e+262:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 100000000.0:
		tmp = 4.0
	elif t_1 <= 1e+158:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(x * Float64(4.0 / y))
	tmp = 0.0
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+158)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -4.0) / y;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = x * (4.0 / y);
	tmp = 0.0;
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+158)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+262], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 100000000.0], 4.0, If[LessEqual[t$95$1, 1e+158], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e262 or 9.99999999999999953e157 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 62.2

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

   8. Simplified 62.2%

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

   if -2e262 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999953e157

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 57.3

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

   5. Simplified 57.3%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-frac-neg2 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 57.0

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

   7. Applied egg-rr 57.0%

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

   if -500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 95.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -500:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 10^{+158}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := x \cdot \frac{4}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+271}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (* x (/ 4.0 y))))
   (if (<= t_1 -2e+262)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 100000000.0) 4.0 (if (<= t_1 1e+271) t_2 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+271) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = x * (4.0d0 / y)
    if (t_1 <= (-2d+262)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 100000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 1d+271) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 100000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+271) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = x * (4.0 / y)
	tmp = 0
	if t_1 <= -2e+262:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 100000000.0:
		tmp = 4.0
	elif t_1 <= 1e+271:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(x * Float64(4.0 / y))
	tmp = 0.0
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+271)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = x * (4.0 / y);
	tmp = 0.0;
	if (t_1 <= -2e+262)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 100000000.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+271)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+262], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 100000000.0], 4.0, If[LessEqual[t$95$1, 1e+271], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e262 or 9.99999999999999953e270 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

         \[\leadsto z \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{4}}{y}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto z \cdot \color{blue}{\frac{\mathsf{neg}\left(4\right)}{y}} \]

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 64.8

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

   5. Simplified 64.8%

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

   if -2e262 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999953e270

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 56.4

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

   5. Simplified 56.4%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-frac-neg2 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 56.1

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

   7. Applied egg-rr 56.1%

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

   if -500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 95.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -500:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 100000000:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 10^{+271}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - z\right)}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -4000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x z)) y)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -4000000000.0)
     t_0
     (if (<= t_1 100000000.0) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - z)) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 100000000.0) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - z)) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -4000000000.0)
		tmp = t_0;
	elseif (t_1 <= 100000000.0)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000000000.0], t$95$0, If[LessEqual[t$95$1, 100000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -4e9 or 1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   5. Simplified 99.4%

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

   if -4e9 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 99.8

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

   8. Simplified 99.8%

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

Alternative 6: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - z\right) \cdot \frac{4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -4000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x z) (/ 4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -4000000000.0)
     t_0
     (if (<= t_1 100000000.0) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 100000000.0) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) * Float64(4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -4000000000.0)
		tmp = t_0;
	elseif (t_1 <= 100000000.0)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000000000.0], t$95$0, If[LessEqual[t$95$1, 100000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -4e9 or 1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   5. Simplified 99.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if -4e9 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 99.8

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

   8. Simplified 99.8%

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

Alternative 7: 67.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -500.0) t_0 (if (<= t_1 5.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-500.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 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 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

         \[\leadsto z \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{4}}{y}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto z \cdot \color{blue}{\frac{\mathsf{neg}\left(4\right)}{y}} \]

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 51.0

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

   5. Simplified 51.0%

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

   if -500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 96.6%

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

Alternative 8: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x y) 4.0 4.0)))
   (if (<= x -6e+33) t_0 (if (<= x 8e+44) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / y), 4.0, 4.0);
	double tmp;
	if (x <= -6e+33) {
		tmp = t_0;
	} else if (x <= 8e+44) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / y), 4.0, 4.0)
	tmp = 0.0
	if (x <= -6e+33)
		tmp = t_0;
	elseif (x <= 8e+44)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]}, If[LessEqual[x, -6e+33], t$95$0, If[LessEqual[x, 8e+44], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.99999999999999967e33 or 8.0000000000000007e44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 83.6%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \frac{-4}{y}, -4\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. distribute-frac-neg2 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. /-lowering-/.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   if -5.99999999999999967e33 < x < 8.0000000000000007e44

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 92.2

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

   8. Simplified 92.2%

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

Alternative 9: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) y)))
   (if (<= x -3.6e+151) t_0 (if (<= x 9e+44) (fma -4.0 (/ 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 <= -3.6e+151) {
		tmp = t_0;
	} else if (x <= 9e+44) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (x <= -3.6e+151)
		tmp = t_0;
	elseif (x <= 9e+44)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -3.6e+151], t$95$0, If[LessEqual[x, 9e+44], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e151 or 9e44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   if -3.6e151 < x < 9e44

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 87.4

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

   8. Simplified 87.4%

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

Alternative 10: 34.6% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

5. Simplified 32.2%

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

Reproduce

herbie shell --seed 2024200 
(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)))