Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.7% → 96.7%

Time: 13.4s

Alternatives: 9

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.9021185125834612e-208
  2. y = -4.678127582372616e-230
  3. z = 9.911447186153516e-16
  4. t = 7.630053886535879e+127
  5. a = -9.206528745099727e+201
  6. b = 5.573814750340383e+86

• 40.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\log z - t\right) \cdot y + \left(\log \left(1 - z\right) - b\right) \cdot a} \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (exp (+ (* (- (log z) t) y) (* (- (log (- 1.0 z)) b) a))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return exp((((log(z) - t) * y) + ((log((1.0 - z)) - b) * a))) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp((((log(z) - t) * y) + ((log((1.0d0 - z)) - b) * a))) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.exp((((Math.log(z) - t) * y) + ((Math.log((1.0 - z)) - b) * a))) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.exp((((math.log(z) - t) * y) + ((math.log((1.0 - z)) - b) * a))) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(exp(Float64(Float64(Float64(log(z) - t) * y) + Float64(Float64(log(Float64(1.0 - z)) - b) * a))) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = exp((((log(z) - t) * y) + ((log((1.0 - z)) - b) * a))) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Exp[N[(N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Final simplification 96.8%

   \[\leadsto e^{\left(\log z - t\right) \cdot y + \left(\log \left(1 - z\right) - b\right) \cdot a} \cdot x \]
4. Add Preprocessing

Alternative 2: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -1.32 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;{\left(e^{2}\right)}^{\left(0.5 \cdot \left(\left(\log z - t\right) \cdot y\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (- z) b) a)) x)))
   (if (<= a -1.32e+116)
     t_1
     (if (<= a 5.5e+139)
       (* (pow (exp 2.0) (* 0.5 (* (- (log z) t) y))) x)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -1.32e+116) {
		tmp = t_1;
	} else if (a <= 5.5e+139) {
		tmp = pow(exp(2.0), (0.5 * ((log(z) - t) * y))) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((-z - b) * a)) * x
    if (a <= (-1.32d+116)) then
        tmp = t_1
    else if (a <= 5.5d+139) then
        tmp = (exp(2.0d0) ** (0.5d0 * ((log(z) - t) * y))) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -1.32e+116) {
		tmp = t_1;
	} else if (a <= 5.5e+139) {
		tmp = Math.pow(Math.exp(2.0), (0.5 * ((Math.log(z) - t) * y))) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((-z - b) * a)) * x
	tmp = 0
	if a <= -1.32e+116:
		tmp = t_1
	elif a <= 5.5e+139:
		tmp = math.pow(math.exp(2.0), (0.5 * ((math.log(z) - t) * y))) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x)
	tmp = 0.0
	if (a <= -1.32e+116)
		tmp = t_1;
	elseif (a <= 5.5e+139)
		tmp = Float64((exp(2.0) ^ Float64(0.5 * Float64(Float64(log(z) - t) * y))) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((-z - b) * a)) * x;
	tmp = 0.0;
	if (a <= -1.32e+116)
		tmp = t_1;
	elseif (a <= 5.5e+139)
		tmp = (exp(2.0) ^ (0.5 * ((log(z) - t) * y))) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -1.32e+116], t$95$1, If[LessEqual[a, 5.5e+139], N[(N[Power[N[Exp[2.0], $MachinePrecision], N[(0.5 * N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.32000000000000002e116 or 5.4999999999999996e139 < a

   1. Initial program 91.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 92.4

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 92.4%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   if -1.32000000000000002e116 < a < 5.4999999999999996e139

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 83.4

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 83.4%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.4%

      \[\leadsto x \cdot {\left({\left({\left(\frac{z}{e^{t}}\right)}^{y}\right)}^{0.5}\right)}^{\color{blue}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.0%

      \[\leadsto x \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\left(\left(\log z - t\right) \cdot y\right) \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+116}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;{\left(e^{2}\right)}^{\left(0.5 \cdot \left(\left(\log z - t\right) \cdot y\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -1.32 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (- z) b) a)) x)))
   (if (<= a -1.32e+116)
     t_1
     (if (<= a 5.5e+139) (* (exp (* (- (log z) t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -1.32e+116) {
		tmp = t_1;
	} else if (a <= 5.5e+139) {
		tmp = exp(((log(z) - t) * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((-z - b) * a)) * x
    if (a <= (-1.32d+116)) then
        tmp = t_1
    else if (a <= 5.5d+139) then
        tmp = exp(((log(z) - t) * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -1.32e+116) {
		tmp = t_1;
	} else if (a <= 5.5e+139) {
		tmp = Math.exp(((Math.log(z) - t) * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((-z - b) * a)) * x
	tmp = 0
	if a <= -1.32e+116:
		tmp = t_1
	elif a <= 5.5e+139:
		tmp = math.exp(((math.log(z) - t) * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x)
	tmp = 0.0
	if (a <= -1.32e+116)
		tmp = t_1;
	elseif (a <= 5.5e+139)
		tmp = Float64(exp(Float64(Float64(log(z) - t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((-z - b) * a)) * x;
	tmp = 0.0;
	if (a <= -1.32e+116)
		tmp = t_1;
	elseif (a <= 5.5e+139)
		tmp = exp(((log(z) - t) * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -1.32e+116], t$95$1, If[LessEqual[a, 5.5e+139], N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.32000000000000002e116 or 5.4999999999999996e139 < a

   1. Initial program 91.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 92.4

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 92.4%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   if -1.32000000000000002e116 < a < 5.4999999999999996e139

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right)} \cdot y} \]

      4. lower-log.f64 91.0

         \[\leadsto x \cdot e^{\left(\color{blue}{\log z} - t\right) \cdot y} \]

   5. Applied rewrites 91.0%

      \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+116}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -0.88:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (- z) b) a)) x)))
   (if (<= a -0.88) t_1 (if (<= a 6.8e-17) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -0.88) {
		tmp = t_1;
	} else if (a <= 6.8e-17) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((-z - b) * a)) * x
    if (a <= (-0.88d0)) then
        tmp = t_1
    else if (a <= 6.8d-17) then
        tmp = exp((-t * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((-z - b) * a)) * x;
	double tmp;
	if (a <= -0.88) {
		tmp = t_1;
	} else if (a <= 6.8e-17) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((-z - b) * a)) * x
	tmp = 0
	if a <= -0.88:
		tmp = t_1
	elif a <= 6.8e-17:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x)
	tmp = 0.0
	if (a <= -0.88)
		tmp = t_1;
	elseif (a <= 6.8e-17)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((-z - b) * a)) * x;
	tmp = 0.0;
	if (a <= -0.88)
		tmp = t_1;
	elseif (a <= 6.8e-17)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -0.88], t$95$1, If[LessEqual[a, 6.8e-17], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 6.7999999999999996e-17 < a

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 81.1

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   if -0.880000000000000004 < a < 6.7999999999999996e-17

   1. Initial program 99.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot t\right) \cdot y}} \]

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 76.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 76.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -0.88:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= a -0.88) t_1 (if (<= a 6.8e-17) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-b * a)) * x;
	double tmp;
	if (a <= -0.88) {
		tmp = t_1;
	} else if (a <= 6.8e-17) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-b * a)) * x
    if (a <= (-0.88d0)) then
        tmp = t_1
    else if (a <= 6.8d-17) then
        tmp = exp((-t * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-b * a)) * x;
	double tmp;
	if (a <= -0.88) {
		tmp = t_1;
	} else if (a <= 6.8e-17) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-b * a)) * x
	tmp = 0
	if a <= -0.88:
		tmp = t_1
	elif a <= 6.8e-17:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-b) * a)) * x)
	tmp = 0.0
	if (a <= -0.88)
		tmp = t_1;
	elseif (a <= 6.8e-17)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-b * a)) * x;
	tmp = 0.0;
	if (a <= -0.88)
		tmp = t_1;
	elseif (a <= 6.8e-17)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -0.88], t$95$1, If[LessEqual[a, 6.8e-17], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 6.7999999999999996e-17 < a

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      5. lower-neg.f64 74.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-b\right)} \cdot a} \]

   5. Applied rewrites 74.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-b\right) \cdot a}} \]

   if -0.880000000000000004 < a < 6.7999999999999996e-17

   1. Initial program 99.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot t\right) \cdot y}} \]

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 76.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 76.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.88:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;a \leq -0.0011:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= a -0.0011) t_1 (if (<= a 5.2e-14) (* (pow z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-b * a)) * x;
	double tmp;
	if (a <= -0.0011) {
		tmp = t_1;
	} else if (a <= 5.2e-14) {
		tmp = pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-b * a)) * x
    if (a <= (-0.0011d0)) then
        tmp = t_1
    else if (a <= 5.2d-14) then
        tmp = (z ** y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-b * a)) * x;
	double tmp;
	if (a <= -0.0011) {
		tmp = t_1;
	} else if (a <= 5.2e-14) {
		tmp = Math.pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-b * a)) * x
	tmp = 0
	if a <= -0.0011:
		tmp = t_1
	elif a <= 5.2e-14:
		tmp = math.pow(z, y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-b) * a)) * x)
	tmp = 0.0
	if (a <= -0.0011)
		tmp = t_1;
	elseif (a <= 5.2e-14)
		tmp = Float64((z ^ y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-b * a)) * x;
	tmp = 0.0;
	if (a <= -0.0011)
		tmp = t_1;
	elseif (a <= 5.2e-14)
		tmp = (z ^ y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -0.0011], t$95$1, If[LessEqual[a, 5.2e-14], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00110000000000000007 or 5.19999999999999993e-14 < a

   1. Initial program 94.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      5. lower-neg.f64 74.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-b\right)} \cdot a} \]

   5. Applied rewrites 74.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-b\right) \cdot a}} \]

   if -0.00110000000000000007 < a < 5.19999999999999993e-14

   1. Initial program 99.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 86.6

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 86.6%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 70.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0011:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-a\right) \cdot z} \cdot x\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+128}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- a) z)) x)))
   (if (<= a -5.2e+99) t_1 (if (<= a 2.3e+128) (* (pow z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-a * z)) * x;
	double tmp;
	if (a <= -5.2e+99) {
		tmp = t_1;
	} else if (a <= 2.3e+128) {
		tmp = pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-a * z)) * x
    if (a <= (-5.2d+99)) then
        tmp = t_1
    else if (a <= 2.3d+128) then
        tmp = (z ** y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-a * z)) * x;
	double tmp;
	if (a <= -5.2e+99) {
		tmp = t_1;
	} else if (a <= 2.3e+128) {
		tmp = Math.pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-a * z)) * x
	tmp = 0
	if a <= -5.2e+99:
		tmp = t_1
	elif a <= 2.3e+128:
		tmp = math.pow(z, y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-a) * z)) * x)
	tmp = 0.0
	if (a <= -5.2e+99)
		tmp = t_1;
	elseif (a <= 2.3e+128)
		tmp = Float64((z ^ y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-a * z)) * x;
	tmp = 0.0;
	if (a <= -5.2e+99)
		tmp = t_1;
	elseif (a <= 2.3e+128)
		tmp = (z ^ y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-a) * z), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -5.2e+99], t$95$1, If[LessEqual[a, 2.3e+128], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.1999999999999999e99 or 2.29999999999999998e128 < a

   1. Initial program 91.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 90.4

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 90.4%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.4%

      \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{\left(z + b\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot \color{blue}{z}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.0%

       \[\leadsto x \cdot e^{\left(-a\right) \cdot z} \]

   if -5.1999999999999999e99 < a < 2.29999999999999998e128

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 82.9

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 82.9%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 65.3%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+99}:\\ \;\;\;\;e^{\left(-a\right) \cdot z} \cdot x\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+128}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-a\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ {z}^{y} \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (pow z y) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return pow(z, y) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (z ** y) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.pow(z, y) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.pow(z, y) * x

Julia

function code(x, y, z, t, a, b)
	return Float64((z ^ y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (z ^ y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

   2. exp-prod N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   3. lower-pow.f64 N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   4. exp-diff N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

   5. rem-exp-log N/A

      \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

   6. lower-/.f64 N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

   7. lower-exp.f64 70.1

      \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

5. Applied rewrites 70.1%

   \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

6. Taylor expanded in t around 0

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

8. Applied rewrites 51.4%

   \[\leadsto x \cdot {z}^{\color{blue}{y}} \]

9. Final simplification 51.4%

   \[\leadsto {z}^{y} \cdot x \]
10. Add Preprocessing

Alternative 9: 19.2% accurate, 54.7× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

   2. exp-prod N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   3. lower-pow.f64 N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   4. exp-diff N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

   5. rem-exp-log N/A

      \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

   6. lower-/.f64 N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

   7. lower-exp.f64 70.1

      \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

5. Applied rewrites 70.1%

   \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

6. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 18.3%

   \[\leadsto x \cdot 1 \]

9. Final simplification 18.3%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))