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

Percentage Accurate: 99.8% → 100.0%

Time: 8.8s

Alternatives: 9

Speedup: 1.5×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ t_2 := \frac{z}{y \cdot -0.25}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y))
        (t_2 (/ z (* y -0.25))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e+16)
       t_2
       (if (<= t_1 2.0)
         2.0
         (if (<= t_1 5e+95) (* z (/ -4.0 y)) (if (<= t_1 5e+267) t_0 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_2 = z / (y * -0.25);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+95) {
		tmp = z * (-4.0 / y);
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_2 = z / (y * -0.25);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+95) {
		tmp = z * (-4.0 / y);
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (y * 0.25)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	t_2 = z / (y * -0.25)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= -1e+16:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 2.0
	elif t_1 <= 5e+95:
		tmp = z * (-4.0 / y)
	elif t_1 <= 5e+267:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(y * 0.25))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	t_2 = Float64(z / Float64(y * -0.25))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+95)
		tmp = Float64(z * Float64(-4.0 / y));
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (y * 0.25);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	t_2 = z / (y * -0.25);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+95)
		tmp = z * (-4.0 / y);
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(z / N[(y * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e+16], t$95$2, If[LessEqual[t$95$1, 2.0], 2.0, If[LessEqual[t$95$1, 5e+95], N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -inf.0 or 5.00000000000000025e95 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 4.9999999999999999e267

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 68.9

         \[\leadsto \frac{x}{\color{blue}{0.25 \cdot y}} \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{x}{0.25 \cdot y}} \]

   if -inf.0 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1e16 or 4.9999999999999999e267 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 97.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 66.1

         \[\leadsto \frac{z}{y \cdot \color{blue}{-0.25}} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{z}{y \cdot -0.25}} \]

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 2

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.8%

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

   if 2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5.00000000000000025e95

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.4

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

   7. Applied rewrites 69.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -\infty:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{y \cdot -0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 2:\\ \;\;\;\;2\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ t_2 := z \cdot \frac{-4}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y))
        (t_2 (* z (/ -4.0 y))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e+16)
       t_2
       (if (<= t_1 2.0)
         2.0
         (if (<= t_1 5e+95) t_2 (if (<= t_1 5e+267) t_0 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_2 = z * (-4.0 / y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double t_2 = z * (-4.0 / y);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= -1e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else if (t_1 <= 5e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+267) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (y * 0.25)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	t_2 = z * (-4.0 / y)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0
	elif t_1 <= -1e+16:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 2.0
	elif t_1 <= 5e+95:
		tmp = t_2
	elif t_1 <= 5e+267:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(y * 0.25))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	t_2 = Float64(z * Float64(-4.0 / y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (y * 0.25);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	t_2 = z * (-4.0 / y);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0;
	elseif (t_1 <= -1e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	elseif (t_1 <= 5e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+267)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e+16], t$95$2, If[LessEqual[t$95$1, 2.0], 2.0, If[LessEqual[t$95$1, 5e+95], t$95$2, If[LessEqual[t$95$1, 5e+267], 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 1/4 binary64))) z)) y) < -inf.0 or 5.00000000000000025e95 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 4.9999999999999999e267

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 68.9

         \[\leadsto \frac{x}{\color{blue}{0.25 \cdot y}} \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{x}{0.25 \cdot y}} \]

   if -inf.0 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < -1e16 or 2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5.00000000000000025e95 or 4.9999999999999999e267 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 98.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 66.5

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

   7. Applied rewrites 66.5%

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

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 2

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -\infty:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 2:\\ \;\;\;\;2\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y \cdot 0.25}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x z) (* y 0.25)))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -2e+19)
     t_0
     (if (<= t_1 50000000000.0) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) / (y * 0.25);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = t_0;
	} else if (t_1 <= 50000000000.0) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) / Float64(y * 0.25))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = t_0;
	elseif (t_1 <= 50000000000.0)
		tmp = fma(-4.0, Float64(z / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], t$95$0, If[LessEqual[t$95$1, 50000000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $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 1/4 binary64))) z)) y) < -2e19 or 5e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 99.7

          \[\leadsto \frac{x - z}{\color{blue}{0.25 \cdot y}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x - z}{0.25 \cdot y}} \]

   if -2e19 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5e10

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-inverses N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

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

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

      12. associate-*l* N/A

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

      13. associate-+r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2} + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right)} \]

      16. associate-*l* N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(\frac{1}{y} \cdot z\right)}\right)\right) \]

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

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(\frac{1}{y} \cdot z\right)} \]

      18. associate-*l/ N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\frac{1 \cdot z}{y}} \]

      19. *-lft-identity N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{\color{blue}{z}}{y} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{2 + \frac{z \cdot -4}{y}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x - z}{y \cdot 0.25}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - z\right) \cdot \frac{4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x z) (/ 4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -2e+19)
     t_0
     (if (<= t_1 50000000000.0) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = t_0;
	} else if (t_1 <= 50000000000.0) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) * Float64(4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = t_0;
	elseif (t_1 <= 50000000000.0)
		tmp = fma(-4.0, Float64(z / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], t$95$0, If[LessEqual[t$95$1, 50000000000.0], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $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 1/4 binary64))) z)) y) < -2e19 or 5e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 99.7

          \[\leadsto \frac{x - z}{\color{blue}{0.25 \cdot y}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x - z}{0.25 \cdot y}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -2e19 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 5e10

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-inverses N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

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

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

      12. associate-*l* N/A

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

      13. associate-+r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2} + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right)} \]

      16. associate-*l* N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(\frac{1}{y} \cdot z\right)}\right)\right) \]

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

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(\frac{1}{y} \cdot z\right)} \]

      18. associate-*l/ N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\frac{1 \cdot z}{y}} \]

      19. *-lft-identity N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{\color{blue}{z}}{y} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{2 + \frac{z \cdot -4}{y}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
   (if (<= t_1 -1e+16) t_0 (if (<= t_1 2.0) 2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -1e+16) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * ((x + (y * 0.25d0)) - z)) / y
    if (t_1 <= (-1d+16)) then
        tmp = t_0
    else if (t_1 <= 2.0d0) then
        tmp = 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	double tmp;
	if (t_1 <= -1e+16) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y
	tmp = 0
	if t_1 <= -1e+16:
		tmp = t_0
	elif t_1 <= 2.0:
		tmp = 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -1e+16)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.25)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -1e+16)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+16], t$95$0, If[LessEqual[t$95$1, 2.0], 2.0, 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 1/4 binary64))) z)) y) < -1e16 or 2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y)

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 54.8

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

   7. Applied rewrites 54.8%

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

   if -1e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 1/4 binary64))) z)) y) < 2

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \leq 2:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -4.0 (/ z y) 2.0)))
   (if (<= z -6.5e-27) t_0 (if (<= z 0.16) (fma 4.0 (/ x y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-4.0, (z / y), 2.0);
	double tmp;
	if (z <= -6.5e-27) {
		tmp = t_0;
	} else if (z <= 0.16) {
		tmp = fma(4.0, (x / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-4.0, Float64(z / y), 2.0)
	tmp = 0.0
	if (z <= -6.5e-27)
		tmp = t_0;
	elseif (z <= 0.16)
		tmp = fma(4.0, Float64(x / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[z, -6.5e-27], t$95$0, If[LessEqual[z, 0.16], N[(4.0 * N[(x / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000025e-27 or 0.160000000000000003 < z

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-inverses N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

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

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

      12. associate-*l* N/A

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

      13. associate-+r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2} + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right)} \]

      16. associate-*l* N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(\frac{1}{y} \cdot z\right)}\right)\right) \]

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

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(\frac{1}{y} \cdot z\right)} \]

      18. associate-*l/ N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\frac{1 \cdot z}{y}} \]

      19. *-lft-identity N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{\color{blue}{z}}{y} \]

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{2 + \frac{z \cdot -4}{y}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if -6.50000000000000025e-27 < z < 0.160000000000000003

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 94.1

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

   8. Applied rewrites 94.1%

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

Alternative 8: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 0.25}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (* y 0.25))))
   (if (<= x -1.1e+141) t_0 (if (<= x 6.5e+141) (fma -4.0 (/ z y) 2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x / (y * 0.25);
	double tmp;
	if (x <= -1.1e+141) {
		tmp = t_0;
	} else if (x <= 6.5e+141) {
		tmp = fma(-4.0, (z / y), 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(y * 0.25))
	tmp = 0.0
	if (x <= -1.1e+141)
		tmp = t_0;
	elseif (x <= 6.5e+141)
		tmp = fma(-4.0, Float64(z / y), 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(y * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+141], t$95$0, If[LessEqual[x, 6.5e+141], N[(-4.0 * N[(z / y), $MachinePrecision] + 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e141 or 6.50000000000000053e141 < x

   1. Initial program 97.5%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 81.0

         \[\leadsto \frac{x}{\color{blue}{0.25 \cdot y}} \]

   5. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{x}{0.25 \cdot y}} \]

   if -1.1e141 < x < 6.50000000000000053e141

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-inverses N/A

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

      9. *-lft-identity N/A

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

      10. associate-*l/ N/A

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

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

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

      12. associate-*l* N/A

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

      13. associate-+r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2} + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right)} \]

      16. associate-*l* N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(\frac{1}{y} \cdot z\right)}\right)\right) \]

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

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(\frac{1}{y} \cdot z\right)} \]

      18. associate-*l/ N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\frac{1 \cdot z}{y}} \]

      19. *-lft-identity N/A

          \[\leadsto 2 + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{\color{blue}{z}}{y} \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{2 + \frac{z \cdot -4}{y}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.4% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 2.0)

C

double code(double x, double y, double z) {
	return 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 2.0;
}

Python

def code(x, y, z):
	return 2.0

Julia

function code(x, y, z)
	return 2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0;
end

Wolfram

code[x_, y_, z_] := 2.0

Derivation

1. Initial program 99.2%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{2} \]
4. Step-by-step derivation

5. Applied rewrites 35.9%

   \[\leadsto \color{blue}{2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))