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

Percentage Accurate: 99.8% → 100.0%

Time: 7.8s

Alternatives: 13

Speedup: 1.5×

• 21.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. *-inverses N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 67.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot z}{y}\\ t_1 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 40000000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 z) y))
        (t_1 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (<= t_1 -20000.0)
     t_0
     (if (<= t_1 40000000000.0)
       4.0
       (if (<= t_1 2e+112) (* (/ x y) 4.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * z) / y;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = (x / 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) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) * z) / y
    t_1 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 40000000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+112) then
        tmp = (x / 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 * z) / y;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 * z) / y
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 40000000000.0:
		tmp = 4.0
	elif t_1 <= 2e+112:
		tmp = (x / y) * 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * z) / y)
	t_1 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 * z) / y;
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = (x / y) * 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 40000000000.0], 4.0, If[LessEqual[t$95$1, 2e+112], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 1.9999999999999999e112 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.3%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 4e10

   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. Applied rewrites 89.5%

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

   if 4e10 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 1.9999999999999999e112

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

Alternative 3: 67.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 40000000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) z))
        (t_1 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (<= t_1 -20000.0)
     t_0
     (if (<= t_1 40000000000.0)
       4.0
       (if (<= t_1 2e+112) (* (/ x y) 4.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * z;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = (x / 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) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) / y) * z
    t_1 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 40000000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+112) then
        tmp = (x / 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 / y) * z;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 / y) * z
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 40000000000.0:
		tmp = 4.0
	elif t_1 <= 2e+112:
		tmp = (x / y) * 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 / y) * z)
	t_1 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 / y) * z;
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = (x / y) * 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 40000000000.0], 4.0, If[LessEqual[t$95$1, 2e+112], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 1.9999999999999999e112 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 4e10

   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. Applied rewrites 89.5%

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

   if 4e10 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 1.9999999999999999e112

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

Alternative 4: 67.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 40000000000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) z))
        (t_1 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (<= t_1 -20000.0)
     t_0
     (if (<= t_1 40000000000.0)
       4.0
       (if (<= t_1 2e+112) (* x (/ 4.0 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * z;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = x * (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) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) / y) * z
    t_1 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 40000000000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+112) then
        tmp = x * (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 / y) * z;
	double t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 40000000000.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+112) {
		tmp = x * (4.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 / y) * z
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 40000000000.0:
		tmp = 4.0
	elif t_1 <= 2e+112:
		tmp = x * (4.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 / y) * z)
	t_1 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = Float64(x * Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 / y) * z;
	t_1 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 40000000000.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+112)
		tmp = x * (4.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 40000000000.0], 4.0, If[LessEqual[t$95$1, 2e+112], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 1.9999999999999999e112 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 4e10

   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. Applied rewrites 89.5%

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

   if 4e10 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 1.9999999999999999e112

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   7. Applied rewrites 64.1%

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

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 5\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 5.0)))
     (* (/ (- x z) y) 4.0)
     (fma (/ x y) 4.0 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 5.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 5.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 5.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 5 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.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 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r/ N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 97.9

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

   8. Applied rewrites 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -20000 \lor \neg \left(1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 5\right):\\ \;\;\;\;\frac{4}{y} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 5.0)))
     (* (/ 4.0 y) (- x z))
     (fma (/ x y) 4.0 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 5.0)) {
		tmp = (4.0 / y) * (x - z);
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 5.0))
		tmp = Float64(Float64(4.0 / y) * Float64(x - z));
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 5.0]], $MachinePrecision]], N[(N[(4.0 / y), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 5 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.7%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.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 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r/ N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 97.9

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

   8. Applied rewrites 97.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -20000 \lor \neg \left(1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5\right):\\ \;\;\;\;\frac{4}{y} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 40000000000\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 40000000000.0)))
     (* x (/ 4.0 y))
     4.0)))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 40000000000.0)) {
		tmp = x * (4.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 40000000000.0d0))) then
        tmp = x * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 40000000000.0)) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 40000000000.0):
		tmp = x * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 40000000000.0))
		tmp = Float64(x * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 40000000000.0)))
		tmp = x * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 40000000000.0]], $MachinePrecision]], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < -2e4 or 4e10 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y))

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 50.7

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 50.6%

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

   if -2e4 < (+.f64 #s(literal 1 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)) < 4e10

   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. Applied rewrites 89.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -20000 \lor \neg \left(1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 40000000000\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-17} \lor \neg \left(z \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.1e-17) (not (<= z 6.5e+67)))
   (fma (/ z y) -4.0 4.0)
   (fma (/ x y) 4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.1e-17) || !(z <= 6.5e+67)) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.1e-17) || !(z <= 6.5e+67))
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.1e-17], N[Not[LessEqual[z, 6.5e+67]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0999999999999998e-17 or 6.4999999999999995e67 < z

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.1%

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

   if -3.0999999999999998e-17 < z < 6.4999999999999995e67

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r/ N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 88.3

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

   8. Applied rewrites 88.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-17} \lor \neg \left(z \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-17} \lor \neg \left(z \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.1e-17) (not (<= z 6.5e+67)))
   (fma (/ z y) -4.0 4.0)
   (fma (/ 4.0 y) x 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.1e-17) || !(z <= 6.5e+67)) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = fma((4.0 / y), x, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.1e-17) || !(z <= 6.5e+67))
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = fma(Float64(4.0 / y), x, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.1e-17], N[Not[LessEqual[z, 6.5e+67]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0999999999999998e-17 or 6.4999999999999995e67 < z

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.1%

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

   if -3.0999999999999998e-17 < z < 6.4999999999999995e67

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-17} \lor \neg \left(z \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+167} \lor \neg \left(x \leq 2.6 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e+167) (not (<= x 2.6e+162)))
   (* (/ x y) 4.0)
   (fma (/ z y) -4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e+167) || !(x <= 2.6e+162)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = fma((z / y), -4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e+167) || !(x <= 2.6e+162))
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = fma(Float64(z / y), -4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e+167], N[Not[LessEqual[x, 2.6e+162]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000012e167 or 2.6e162 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -3.00000000000000012e167 < x < 2.6e162

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+167} \lor \neg \left(x \leq 2.6 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+167} \lor \neg \left(x \leq 2.6 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{y}, z, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e+167) (not (<= x 2.6e+162)))
   (* (/ x y) 4.0)
   (fma (/ -4.0 y) z 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e+167) || !(x <= 2.6e+162)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = fma((-4.0 / y), z, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e+167) || !(x <= 2.6e+162))
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = fma(Float64(-4.0 / y), z, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e+167], N[Not[LessEqual[x, 2.6e+162]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(-4.0 / y), $MachinePrecision] * z + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000012e167 or 2.6e162 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -3.00000000000000012e167 < x < 2.6e162

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. *-inverses N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.0%

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

   10. Applied rewrites 78.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+167} \lor \neg \left(x \leq 2.6 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{y}, z, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. *-inverses N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.8%

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

Alternative 13: 34.3% 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 98.8%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2024339 
(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)))