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

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 15

Speedup: 1.0×

• 58.3% 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: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -1000.0) (exp x) (if (<= t_0 5e+115) (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 <= -1000.0) {
		tmp = exp(x);
	} else if (t_0 <= 5e+115) {
		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 <= (-1000.0d0)) then
        tmp = exp(x)
    else if (t_0 <= 5d+115) 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 <= -1000.0) {
		tmp = Math.exp(x);
	} else if (t_0 <= 5e+115) {
		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 <= -1000.0:
		tmp = math.exp(x)
	elif t_0 <= 5e+115:
		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 <= -1000.0)
		tmp = exp(x);
	elseif (t_0 <= 5e+115)
		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 <= -1000.0)
		tmp = exp(x);
	elseif (t_0 <= 5e+115)
		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, -1000.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+115], 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))) < -1e3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 95.3

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

   8. Simplified 95.3%

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

   if -1e3 < (+.f64 x (*.f64 y (log.f64 y))) < 5.00000000000000008e115

   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. lower-neg.f64 80.2

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

   5. Simplified 80.2%

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

   if 5.00000000000000008e115 < (+.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. lower-*.f64 N/A

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

      6. lower-log.f64 69.9

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

   5. Simplified 69.9%

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

      1. lift-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 69.9

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

   7. Applied egg-rr 69.9%

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

Alternative 3: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \left(z \cdot z\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+113}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* (* z z) 0.5)))
   (if (<= t_0 -1000.0) t_1 (if (<= t_0 1e+113) (+ x 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 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+113) {
		tmp = x + 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 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d+113) then
        tmp = x + 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 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+113) {
		tmp = x + 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 <= -1000.0:
		tmp = t_1
	elif t_0 <= 1e+113:
		tmp = x + 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 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e+113)
		tmp = Float64(x + 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 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e+113)
		tmp = x + 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, -1000.0], t$95$1, If[LessEqual[t$95$0, 1e+113], N[(x + 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1e3 or 1e113 < (-.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. lower-neg.f64 49.8

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

   5. Simplified 49.8%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 25.8

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

   8. Simplified 25.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.f64 30.6

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

   11. Simplified 30.6%

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

   if -1e3 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1e113

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 70.3

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

   8. Simplified 70.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 48.7

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

   11. Simplified 48.7%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -1000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 10^{+113}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 2 \cdot 10^{+160}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 2e+160) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 2e+160) {
		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)) <= 2d+160) 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)) <= 2e+160) {
		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)) <= 2e+160:
		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)) <= 2e+160)
		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)) <= 2e+160)
		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], 2e+160], 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)) < 2.00000000000000001e160

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 91.8

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

   5. Simplified 91.8%

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

   if 2.00000000000000001e160 < (*.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. lower-*.f64 N/A

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

      6. lower-log.f64 90.9

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

   5. Simplified 90.9%

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

      1. lift-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 90.9

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

   7. Applied egg-rr 90.9%

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

Alternative 5: 32.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -500:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) -500.0)
   (* (* z z) 0.5)
   (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= -500.0) {
		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 (Float64(Float64(x + Float64(y * log(y))) - z) <= -500.0)
		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[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], -500.0], 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 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -500

   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. lower-neg.f64 56.1

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

   5. Simplified 56.1%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 2.2

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

   8. Simplified 2.2%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.f64 16.2

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

   11. Simplified 16.2%

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

   if -500 < (-.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. lower-neg.f64 52.0

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

   5. Simplified 52.0%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 42.1

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

   8. Simplified 42.1%

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

   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. lower-*.f64 42.0

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

   11. Simplified 42.0%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -500:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+65}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 86:\\ \;\;\;\;{y}^{y}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e+65) (exp x) (if (<= x 86.0) (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+65) {
		tmp = exp(x);
	} else if (x <= 86.0) {
		tmp = pow(y, y);
	} else {
		tmp = exp(x);
	}
	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 (x <= (-7.6d+65)) then
        tmp = exp(x)
    else if (x <= 86.0d0) then
        tmp = y ** y
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+65) {
		tmp = Math.exp(x);
	} else if (x <= 86.0) {
		tmp = Math.pow(y, y);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.6e+65:
		tmp = math.exp(x)
	elif x <= 86.0:
		tmp = math.pow(y, y)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.6e+65)
		tmp = exp(x);
	elseif (x <= 86.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.6e+65)
		tmp = exp(x);
	elseif (x <= 86.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.6e+65], N[Exp[x], $MachinePrecision], If[LessEqual[x, 86.0], N[Power[y, y], $MachinePrecision], N[Exp[x], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.60000000000000022e65 or 86 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 86.4

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

   8. Simplified 86.4%

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

   if -7.60000000000000022e65 < x < 86

   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. lower-*.f64 N/A

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

      6. lower-log.f64 68.6

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

   5. Simplified 68.6%

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

      1. lift-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 68.6

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

   7. Applied egg-rr 68.6%

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

Alternative 7: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, z \cdot 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}, 1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+67)
   (fma
    (* (* z z) (fma z (* z 0.027777777777777776) -0.25))
    (/ 1.0 (fma z -0.16666666666666666 -0.5))
    (- 1.0 z))
   (exp x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+67) {
		tmp = fma(((z * z) * fma(z, (z * 0.027777777777777776), -0.25)), (1.0 / fma(z, -0.16666666666666666, -0.5)), (1.0 - z));
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+67)
		tmp = fma(Float64(Float64(z * z) * fma(z, Float64(z * 0.027777777777777776), -0.25)), Float64(1.0 / fma(z, -0.16666666666666666, -0.5)), Float64(1.0 - z));
	else
		tmp = exp(x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+67], N[(N[(N[(z * z), $MachinePrecision] * N[(z * N[(z * 0.027777777777777776), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * -0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999983e66

   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. lower-neg.f64 90.5

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

   5. Simplified 90.5%

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

   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. lower-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. lower-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. lower-fma.f64 78.8

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

   8. Simplified 78.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+l+ N/A

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

      6. associate-*r* N/A

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

      7. lift-fma.f64 N/A

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

      8. flip-+ N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied egg-rr 88.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, 0.027777777777777776 \cdot z, -0.25\right), \frac{1}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}, 1 - z\right)} \]

   if -9.99999999999999983e66 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 59.5

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

   8. Simplified 59.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, z \cdot 0.027777777777777776, -0.25\right), \frac{1}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}, 1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, z \cdot 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e-5)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.8e+85)
     (fma (* z (fma z (* z 0.25) -1.0)) (/ 1.0 (fma z 0.5 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 <= -3.1e-5) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.8e+85) {
		tmp = fma((z * fma(z, (z * 0.25), -1.0)), (1.0 / fma(z, 0.5, 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 <= -3.1e-5)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.8e+85)
		tmp = fma(Float64(z * fma(z, Float64(z * 0.25), -1.0)), Float64(1.0 / fma(z, 0.5, 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, -3.1e-5], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+85], N[(N[(z * N[(z * N[(z * 0.25), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $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 < -3.10000000000000014e-5

   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. lower-neg.f64 39.4

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

   5. Simplified 39.4%

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

   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. lower-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. lower-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. lower-fma.f64 16.9

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

   8. Simplified 16.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.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. lower-*.f64 35.2

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

   11. Simplified 35.2%

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

   if -3.10000000000000014e-5 < x < 2.7999999999999999e85

   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. lower-neg.f64 67.4

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

   5. Simplified 67.4%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 42.0

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

   8. Simplified 42.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.25 \cdot z, -1\right) \cdot z, \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]

   if 2.7999999999999999e85 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.9

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   10. 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. lower-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. lower-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. lower-fma.f64 85.9

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

   11. Simplified 85.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, z \cdot 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -460.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.8e+85)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -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 <= -460.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.8e+85) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -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 <= -460.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.8e+85)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -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, -460.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+85], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $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 < -460

   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. lower-neg.f64 35.7

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

   5. Simplified 35.7%

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

   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. lower-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. lower-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. lower-fma.f64 13.2

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

   8. Simplified 13.2%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.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. lower-*.f64 32.7

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

   11. Simplified 32.7%

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

   if -460 < x < 2.7999999999999999e85

   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. lower-neg.f64 68.4

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

   5. Simplified 68.4%

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

   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. lower-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. lower-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. lower-fma.f64 43.5

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

   8. Simplified 43.5%

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

   if 2.7999999999999999e85 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.9

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   10. 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. lower-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. lower-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. lower-fma.f64 85.9

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

   11. Simplified 85.9%

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

Alternative 10: 46.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e-5)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.8e+85)
     (fma z (fma z 0.5 -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 <= -3.1e-5) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.8e+85) {
		tmp = fma(z, fma(z, 0.5, -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 <= -3.1e-5)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.8e+85)
		tmp = fma(z, fma(z, 0.5, -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, -3.1e-5], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+85], N[(z * N[(z * 0.5 + -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 < -3.10000000000000014e-5

   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. lower-neg.f64 39.4

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

   5. Simplified 39.4%

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

   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. lower-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. lower-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. lower-fma.f64 16.9

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

   8. Simplified 16.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.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. lower-*.f64 35.2

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

   11. Simplified 35.2%

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

   if -3.10000000000000014e-5 < x < 2.7999999999999999e85

   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. lower-neg.f64 67.4

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

   5. Simplified 67.4%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 42.0

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

   8. Simplified 42.0%

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

   if 2.7999999999999999e85 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.9

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   10. 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. lower-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. lower-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. lower-fma.f64 85.9

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

   11. Simplified 85.9%

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

Alternative 11: 43.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e-5)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 6.8e+135)
     (fma z (fma z 0.5 -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 <= -3.1e-5) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 6.8e+135) {
		tmp = fma(z, fma(z, 0.5, -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 <= -3.1e-5)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 6.8e+135)
		tmp = fma(z, fma(z, 0.5, -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, -3.1e-5], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e+135], N[(z * N[(z * 0.5 + -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 < -3.10000000000000014e-5

   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. lower-neg.f64 39.4

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

   5. Simplified 39.4%

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

   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. lower-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. lower-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. lower-fma.f64 16.9

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

   8. Simplified 16.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.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. lower-*.f64 35.2

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

   11. Simplified 35.2%

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

   if -3.10000000000000014e-5 < x < 6.80000000000000019e135

   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. lower-neg.f64 65.1

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

   5. Simplified 65.1%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 39.5

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

   8. Simplified 39.5%

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

   if 6.80000000000000019e135 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.6

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

   8. Simplified 91.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-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. lower-fma.f64 81.9

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

   11. Simplified 81.9%

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

Alternative 12: 42.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -460.0)
   (* (* z z) 0.5)
   (if (<= x 6.8e+135)
     (fma z (fma z 0.5 -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 <= -460.0) {
		tmp = (z * z) * 0.5;
	} else if (x <= 6.8e+135) {
		tmp = fma(z, fma(z, 0.5, -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 <= -460.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 6.8e+135)
		tmp = fma(z, fma(z, 0.5, -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, -460.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.8e+135], N[(z * N[(z * 0.5 + -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 < -460

   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. lower-neg.f64 35.7

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

   5. Simplified 35.7%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 10.4

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

   8. Simplified 10.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.f64 25.4

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

   11. Simplified 25.4%

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

   if -460 < x < 6.80000000000000019e135

   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. lower-neg.f64 66.1

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

   5. Simplified 66.1%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 40.5

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

   8. Simplified 40.5%

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

   if 6.80000000000000019e135 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.6

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

   8. Simplified 91.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-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. lower-fma.f64 81.9

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

   11. Simplified 81.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -460.0)
   (* (* z z) 0.5)
   (if (<= x 6.8e+135) (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 <= -460.0) {
		tmp = (z * z) * 0.5;
	} else if (x <= 6.8e+135) {
		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 <= -460.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 6.8e+135)
		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, -460.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.8e+135], 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 < -460

   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. lower-neg.f64 35.7

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

   5. Simplified 35.7%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 10.4

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

   8. Simplified 10.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.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. lower-*.f64 25.4

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

   11. Simplified 25.4%

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

   if -460 < x < 6.80000000000000019e135

   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. lower-neg.f64 66.1

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

   5. Simplified 66.1%

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

   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. lower-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. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 40.5

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

   8. Simplified 40.5%

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

   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. lower-*.f64 40.4

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

   11. Simplified 40.4%

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

   if 6.80000000000000019e135 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 91.6

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

   8. Simplified 91.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-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. lower-fma.f64 81.9

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

   11. Simplified 81.9%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
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 y around 0

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 80.6

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

5. Simplified 80.6%

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

6. Taylor expanded in z around 0

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

   1. lower-exp.f64 55.3

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

8. Simplified 55.3%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 13.7

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

11. Simplified 13.7%

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

12. Final simplification 13.7%

    \[\leadsto x + 1 \]
13. 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 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. lower-neg.f64 53.2

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

5. Simplified 53.2%

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

6. Taylor expanded in z around 0

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

8. Simplified 13.3%

   \[\leadsto \color{blue}{1} \]
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 2024207 
(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)))