Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 15

Speedup: 1.0×

• 58.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+92}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -0.0001) (exp x) (if (<= t_0 1e+92) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = exp(x);
	} else if (t_0 <= 1e+92) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * log(y))
    if (t_0 <= (-0.0001d0)) then
        tmp = exp(x)
    else if (t_0 <= 1d+92) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y * Math.log(y));
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = Math.exp(x);
	} else if (t_0 <= 1e+92) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y * math.log(y))
	tmp = 0
	if t_0 <= -0.0001:
		tmp = math.exp(x)
	elif t_0 <= 1e+92:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = exp(x);
	elseif (t_0 <= 1e+92)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y * log(y));
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = exp(x);
	elseif (t_0 <= 1e+92)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e+92], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -1.00000000000000005e-4

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 87.4%

      \[\leadsto e^{\color{blue}{x}} \]

   if -1.00000000000000005e-4 < (+.f64 x (*.f64 y (log.f64 y))) < 1e92

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 86.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 86.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   if 1e92 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. log-lowering-log.f64 75.5

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 75.5%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      2. exp-to-pow N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      3. pow-lowering-pow.f64 75.5

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 75.5%

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

Alternative 3: 31.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \left(z \cdot z\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* (* z z) 0.5)))
   (if (<= t_0 -1e+37) t_1 (if (<= t_0 5e+112) 1.0 t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = (z * z) * 0.5;
	double tmp;
	if (t_0 <= -1e+37) {
		tmp = t_1;
	} else if (t_0 <= 5e+112) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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 = (x + (y * log(y))) - z
    t_1 = (z * z) * 0.5d0
    if (t_0 <= (-1d+37)) then
        tmp = t_1
    else if (t_0 <= 5d+112) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + (y * Math.log(y))) - z;
	double t_1 = (z * z) * 0.5;
	double tmp;
	if (t_0 <= -1e+37) {
		tmp = t_1;
	} else if (t_0 <= 5e+112) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + (y * math.log(y))) - z
	t_1 = (z * z) * 0.5
	tmp = 0
	if t_0 <= -1e+37:
		tmp = t_1
	elif t_0 <= 5e+112:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(Float64(z * z) * 0.5)
	tmp = 0.0
	if (t_0 <= -1e+37)
		tmp = t_1;
	elseif (t_0 <= 5e+112)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + (y * log(y))) - z;
	t_1 = (z * z) * 0.5;
	tmp = 0.0;
	if (t_0 <= -1e+37)
		tmp = t_1;
	elseif (t_0 <= 5e+112)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+37], t$95$1, If[LessEqual[t$95$0, 5e+112], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -9.99999999999999954e36 or 5e112 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 48.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 48.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 26.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 34.3

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 34.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -9.99999999999999954e36 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5e112

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 63.1%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 50.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 5 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 0:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp (- (+ x (* y (log y))) z)) 0.0)
   (* (* z z) 0.5)
   (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(((x + (y * log(y))) - z)) <= 0.0) {
		tmp = (z * z) * 0.5;
	} else {
		tmp = fma(z, (z * 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(Float64(Float64(x + Float64(y * log(y))) - z)) <= 0.0)
		tmp = Float64(Float64(z * z) * 0.5);
	else
		tmp = fma(z, Float64(z * 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)) < 0.0

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 64.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 64.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 2.3

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 23.5

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 23.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if 0.0 < (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 47.1

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 47.1%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 44.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 44.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 44.1

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{0.5 \cdot z}, 1\right) \]

   11. Simplified 44.1%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{0.5 \cdot z}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 0:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 10^{+161}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 1e+161) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 1e+161) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y * log(y)) <= 1d+161) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * Math.log(y)) <= 1e+161) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * math.log(y)) <= 1e+161:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 1e+161)
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * log(y)) <= 1e+161)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 1e+161], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 1e161

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{x} - z} \]
   4. Step-by-step derivation

   5. Simplified 91.1%

      \[\leadsto e^{\color{blue}{x} - z} \]

   if 1e161 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. log-lowering-log.f64 93.3

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 93.3%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      2. exp-to-pow N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      3. pow-lowering-pow.f64 93.3

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 93.3%

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

Alternative 6: 72.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+63}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 2.45e+63) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45e+63) {
		tmp = exp(x);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.45d+63) then
        tmp = exp(x)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45e+63) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.45e+63:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.45e+63)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.45e+63)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.45e+63], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4499999999999998e63

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 66.8%

      \[\leadsto e^{\color{blue}{x}} \]

   if 2.4499999999999998e63 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. log-lowering-log.f64 84.9

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 84.9%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      2. exp-to-pow N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      3. pow-lowering-pow.f64 84.9

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 84.9%

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

Alternative 7: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+51)
   (fma
    z
    (fma
     (* z (fma (* z (* z z)) -0.004629629629629629 0.125))
     (fma z -1.3333333333333333 4.0)
     -1.0)
    1.0)
   (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+51) {
		tmp = fma(z, fma((z * fma((z * (z * z)), -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+51)
		tmp = fma(z, fma(Float64(z * fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	else
		tmp = exp(x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.8e+51], N[(z * N[(N[(z * N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * N[(z * -1.3333333333333333 + 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.7999999999999997e51

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 98.1

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 98.1%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 87.0

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot z} + -1, 1\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{{\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} \cdot z + -1, 1\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z\right) \cdot \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z, \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}, -1\right)}, 1\right) \]

   10. Applied egg-rr 14.1%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \frac{1}{0.25 + \mathsf{fma}\left(z, 0.08333333333333333, \left(z \cdot z\right) \cdot 0.027777777777777776\right)}, -1\right)}, 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{4 + \frac{-4}{3} \cdot z}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{-4}{3} \cdot z + 4}, -1\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{z \cdot \frac{-4}{3}} + 4, -1\right), 1\right) \]

       3. accelerator-lowering-fma.f64 98.1

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   13. Simplified 98.1%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   if -5.7999999999999997e51 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 56.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+33}:\\ \;\;\;\;t\_0 \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(t\_0, -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= x -4.3e+33)
     (* t_0 -0.16666666666666666)
     (if (<= x 5.5e+102)
       (fma
        z
        (fma
         (* z (fma t_0 -0.004629629629629629 0.125))
         (fma z -1.3333333333333333 4.0)
         -1.0)
        1.0)
       (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double tmp;
	if (x <= -4.3e+33) {
		tmp = t_0 * -0.16666666666666666;
	} else if (x <= 5.5e+102) {
		tmp = fma(z, fma((z * fma(t_0, -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (x <= -4.3e+33)
		tmp = Float64(t_0 * -0.16666666666666666);
	elseif (x <= 5.5e+102)
		tmp = fma(z, fma(Float64(z * fma(t_0, -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e+33], N[(t$95$0 * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 5.5e+102], N[(z * N[(N[(z * N[(t$95$0 * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * N[(z * -1.3333333333333333 + 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.30000000000000028e33

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 45.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 45.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 23.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 23.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot \left(z \cdot z\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{{z}^{2}}\right) \]

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

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot {z}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

       6. *-lowering-*.f64 57.3

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 57.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -4.30000000000000028e33 < x < 5.49999999999999981e102

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 62.4

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 62.4%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 36.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 36.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot z} + -1, 1\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{{\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} \cdot z + -1, 1\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z\right) \cdot \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z, \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}, -1\right)}, 1\right) \]

   10. Applied egg-rr 23.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \frac{1}{0.25 + \mathsf{fma}\left(z, 0.08333333333333333, \left(z \cdot z\right) \cdot 0.027777777777777776\right)}, -1\right)}, 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{4 + \frac{-4}{3} \cdot z}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{-4}{3} \cdot z + 4}, -1\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{z \cdot \frac{-4}{3}} + 4, -1\right), 1\right) \]

       3. accelerator-lowering-fma.f64 43.5

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   13. Simplified 43.5%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   if 5.49999999999999981e102 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. accelerator-lowering-fma.f64 95.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+33}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t\_0 \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(t\_0, -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= x -4e+33)
     (* t_0 -0.16666666666666666)
     (if (<= x 5.9e+102)
       (fma z (fma (* z (fma t_0 -0.004629629629629629 0.125)) 4.0 -1.0) 1.0)
       (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double tmp;
	if (x <= -4e+33) {
		tmp = t_0 * -0.16666666666666666;
	} else if (x <= 5.9e+102) {
		tmp = fma(z, fma((z * fma(t_0, -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (x <= -4e+33)
		tmp = Float64(t_0 * -0.16666666666666666);
	elseif (x <= 5.9e+102)
		tmp = fma(z, fma(Float64(z * fma(t_0, -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+33], N[(t$95$0 * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 5.9e+102], N[(z * N[(N[(z * N[(t$95$0 * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999998e33

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 45.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 45.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 23.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 23.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot \left(z \cdot z\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{{z}^{2}}\right) \]

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

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot {z}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

       6. *-lowering-*.f64 57.3

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 57.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -3.9999999999999998e33 < x < 5.90000000000000005e102

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 62.4

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 62.4%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 36.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 36.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot z} + -1, 1\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{{\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} \cdot z + -1, 1\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z\right) \cdot \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}} + -1, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\left({\left(z \cdot \frac{-1}{6}\right)}^{3} + {\frac{1}{2}}^{3}\right) \cdot z, \frac{1}{\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)}, -1\right)}, 1\right) \]

   10. Applied egg-rr 23.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \frac{1}{0.25 + \mathsf{fma}\left(z, 0.08333333333333333, \left(z \cdot z\right) \cdot 0.027777777777777776\right)}, -1\right)}, 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{4}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 37.5%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{4}, -1\right), 1\right) \]

   if 5.90000000000000005e102 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. accelerator-lowering-fma.f64 95.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+29}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e+29)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= x 1e+103)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e+29) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (x <= 1e+103) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e+29)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (x <= 1e+103)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e+29], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 1e+103], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e29

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 46.4

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 46.4%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 22.9

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 22.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot \left(z \cdot z\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{{z}^{2}}\right) \]

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

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot {z}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

       6. *-lowering-*.f64 56.4

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 56.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -5.2e29 < x < 1e103

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 62.1

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 62.1%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 36.7

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 36.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   if 1e103 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

      7. accelerator-lowering-fma.f64 95.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+29}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+30}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e+30)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= x 3.7e+121)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+30) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (x <= 3.7e+121) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e+30)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (x <= 3.7e+121)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e+30], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 3.7e+121], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6999999999999999e30

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 46.4

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 46.4%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. accelerator-lowering-fma.f64 22.9

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 22.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot \left(z \cdot z\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{{z}^{2}}\right) \]

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

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(z \cdot {z}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

       6. *-lowering-*.f64 56.4

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 56.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -4.6999999999999999e30 < x < 3.70000000000000013e121

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 61.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 61.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 36.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 36.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   if 3.70000000000000013e121 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 97.5%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 85.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+30}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+35}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+35)
   (* (* z z) 0.5)
   (if (<= x 3.7e+121)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+35) {
		tmp = (z * z) * 0.5;
	} else if (x <= 3.7e+121) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e+35)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 3.7e+121)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+35], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.7e+121], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000029e35

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 45.3

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 45.3%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 20.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 49.0

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -4.80000000000000029e35 < x < 3.70000000000000013e121

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 61.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 61.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 36.8

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   if 3.70000000000000013e121 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 97.5%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 85.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+35}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.6e+35)
   (* (* z z) 0.5)
   (if (<= x 3.7e+121) (fma z (* z 0.5) 1.0) (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.6e+35) {
		tmp = (z * z) * 0.5;
	} else if (x <= 3.7e+121) {
		tmp = fma(z, (z * 0.5), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.6e+35)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 3.7e+121)
		tmp = fma(z, Float64(z * 0.5), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.6e+35], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.7e+121], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999996e35

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 45.3

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 45.3%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 20.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 49.0

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -4.5999999999999996e35 < x < 3.70000000000000013e121

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 61.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 61.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 36.8

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 36.8

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{0.5 \cdot z}, 1\right) \]

   11. Simplified 36.8%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{0.5 \cdot z}, 1\right) \]

   if 3.70000000000000013e121 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 97.5%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 85.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 14.6% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 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 = x + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 51.6%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 14.1

      \[\leadsto \color{blue}{1 + x} \]

8. Simplified 14.1%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 14.1%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 15: 14.4% accurate, 212.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 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 = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 51.6%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 13.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x - z\right) + \log y \cdot y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (+ (- x z) (* (log y) y))))

C

double code(double x, double y, double z) {
	return exp(((x - z) + (log(y) * 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 = exp(((x - z) + (log(y) * y)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x - z) + (Math.log(y) * y)));
}

Python

def code(x, y, z):
	return math.exp(((x - z) + (math.log(y) * y)))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x - z) + Float64(log(y) * y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x - z) + (log(y) * y)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (exp (+ (- x z) (* (log y) y))))

  (exp (- (+ x (* y (log y))) z)))