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

Percentage Accurate: 99.7% → 99.7%

Time: 7.6s

Alternatives: 11

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $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. Final simplification 99.9%

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

Alternative 2: 66.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{y}\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ t_2 := \frac{-4 \cdot z}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) y))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y))
        (t_2 (/ (* -4.0 z) y)))
   (if (<= t_1 -1e+113)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 5.0) 4.0 (if (<= t_1 2e+155) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / y;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = (x * 4.0d0) / y
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    t_2 = ((-4.0d0) * z) / y
    if (t_1 <= (-1d+113)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+155) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / y;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / y
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	t_2 = (-4.0 * z) / y
	tmp = 0
	if t_1 <= -1e+113:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 2e+155:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / y)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	t_2 = Float64(Float64(-4.0 * z) / y)
	tmp = 0.0
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+155)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / y;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	t_2 = (-4.0 * z) / y;
	tmp = 0.0;
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+155)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+113], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 2e+155], t$95$0, t$95$2]]]]]]]

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) < -1e113 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2.00000000000000001e155

   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. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 61.8%

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

   if -1e113 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 2.00000000000000001e155 < (/.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. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 60.9%

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

   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. Applied rewrites 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\frac{x \cdot 4}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{y} \cdot x\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ t_2 := \frac{-4 \cdot z}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 y) x))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y))
        (t_2 (/ (* -4.0 z) y)))
   (if (<= t_1 -1e+113)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 5.0) 4.0 (if (<= t_1 2e+155) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (4.0d0 / y) * x
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    t_2 = ((-4.0d0) * z) / y
    if (t_1 <= (-1d+113)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+155) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / y) * x
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	t_2 = (-4.0 * z) / y
	tmp = 0
	if t_1 <= -1e+113:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 2e+155:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / y) * x)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	t_2 = Float64(Float64(-4.0 * z) / y)
	tmp = 0.0
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+155)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / y) * x;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	t_2 = (-4.0 * z) / y;
	tmp = 0.0;
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+155)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+113], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 2e+155], t$95$0, t$95$2]]]]]]]

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) < -1e113 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2.00000000000000001e155

   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. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if -1e113 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 2.00000000000000001e155 < (/.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. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 60.9%

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

   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. Applied rewrites 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{y} \cdot x\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ t_2 := \frac{-4}{y} \cdot z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+204}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 y) x))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y))
        (t_2 (* (/ -4.0 y) z)))
   (if (<= t_1 -1e+113)
     t_0
     (if (<= t_1 -500.0)
       t_2
       (if (<= t_1 5.0) 4.0 (if (<= t_1 1e+204) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 / y) * z;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+204) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (4.0d0 / y) * x
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    t_2 = ((-4.0d0) / y) * z
    if (t_1 <= (-1d+113)) then
        tmp = t_0
    else if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 1d+204) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / y) * x;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 / y) * z;
	double tmp;
	if (t_1 <= -1e+113) {
		tmp = t_0;
	} else if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 1e+204) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / y) * x
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	t_2 = (-4.0 / y) * z
	tmp = 0
	if t_1 <= -1e+113:
		tmp = t_0
	elif t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 1e+204:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / y) * x)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	t_2 = Float64(Float64(-4.0 / y) * z)
	tmp = 0.0
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+204)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / y) * x;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	t_2 = (-4.0 / y) * z;
	tmp = 0.0;
	if (t_1 <= -1e+113)
		tmp = t_0;
	elseif (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 1e+204)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+113], t$95$0, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 1e+204], t$95$0, t$95$2]]]]]]]

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) < -1e113 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999989e203

   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. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -1e113 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -500 or 9.99999999999999989e203 < (/.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. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 61.4

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

   5. Applied rewrites 61.4%

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

   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. Applied rewrites 96.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+204}:\\ \;\;\;\;\frac{4}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - z\right) \cdot 4}{y} + 1\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -10000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ (* (- x z) 4.0) y) 1.0))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -10000000000.0)
     t_0
     (if (<= t_1 5.0) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - z) * 4.0) / y) + 1.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -10000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - z) * 4.0) / y) + 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -10000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(z / y), $MachinePrecision] * -4.0 + 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) < -1e10 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 y around 0

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

      1. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -1e10 < (/.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 x around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -10000000000:\\ \;\;\;\;\frac{\left(x - z\right) \cdot 4}{y} + 1\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot 4}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -10000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- x z) y) 4.0)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -10000000000.0)
     t_0
     (if (<= t_1 5.0) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - z) / y) * 4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -10000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - z) / y) * 4.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -10000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(z / y), $MachinePrecision] * -4.0 + 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) < -1e10 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 y around 0

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

      1. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   7. 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 99.0

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

   8. Applied rewrites 99.0%

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

   if -1e10 < (/.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 x around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -10000000000:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot \left(z - x\right)\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -10000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) (- z x)))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -10000000000.0)
     t_0
     (if (<= t_1 5.0) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * (z - x);
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -10000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 / y) * Float64(z - x))
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -10000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / y), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], N[(N[(z / y), $MachinePrecision] * -4.0 + 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) < -1e10 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 y around 0

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

   5. Applied rewrites 98.8%

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

   if -1e10 < (/.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 x around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -10000000000:\\ \;\;\;\;\frac{-4}{y} \cdot \left(z - x\right)\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot \left(z - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 100000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) z)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -500.0) t_0 (if (<= t_1 100000.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * z;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 100000.0) {
		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 = ((-4.0d0) / y) * z
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-500.0d0)) then
        tmp = t_0
    else if (t_1 <= 100000.0d0) 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 = (-4.0 / y) * z;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 100000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 / y) * z
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_0
	elif t_1 <= 100000.0:
		tmp = 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(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 100000.0)
		tmp = 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 = ((((0.75 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 100000.0)
		tmp = 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[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$0, If[LessEqual[t$95$1, 100000.0], 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 1e5 < (/.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. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if -500 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e5

   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 95.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 100000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ z y) -4.0 4.0)))
   (if (<= z -1.26e+76) t_0 (if (<= z 2.1e+107) (fma (/ x y) 4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z / y), -4.0, 4.0);
	double tmp;
	if (z <= -1.26e+76) {
		tmp = t_0;
	} else if (z <= 2.1e+107) {
		tmp = fma((x / y), 4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z / y), -4.0, 4.0)
	tmp = 0.0
	if (z <= -1.26e+76)
		tmp = t_0;
	elseif (z <= 2.1e+107)
		tmp = fma(Float64(x / y), 4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision]}, If[LessEqual[z, -1.26e+76], t$95$0, If[LessEqual[z, 2.1e+107], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26000000000000007e76 or 2.1e107 < z

   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

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-+l+ N/A

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

   5. Applied rewrites 93.2%

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

   if -1.26000000000000007e76 < z < 2.1e107

   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 z around 0

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

   5. Applied rewrites 88.0%

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

   7. Applied rewrites 88.1%

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

Alternative 10: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot z}{y}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 z) y)))
   (if (<= z -1.75e+76) t_0 (if (<= z 1.65e+122) (fma (/ x y) 4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * z) / y;
	double tmp;
	if (z <= -1.75e+76) {
		tmp = t_0;
	} else if (z <= 1.65e+122) {
		tmp = fma((x / y), 4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * z) / y)
	tmp = 0.0
	if (z <= -1.75e+76)
		tmp = t_0;
	elseif (z <= 1.65e+122)
		tmp = fma(Float64(x / y), 4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -1.75e+76], t$95$0, If[LessEqual[z, 1.65e+122], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e76 or 1.6499999999999999e122 < z

   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 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. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 76.0%

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

   if -1.75e76 < z < 1.6499999999999999e122

   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 z around 0

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

Alternative 11: 35.0% 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. Applied rewrites 34.5%

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

Reproduce

herbie shell --seed 2024242 
(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)))