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

Percentage Accurate: 96.5% → 99.7%

Time: 16.2s

Alternatives: 15

Speedup: 1.5×

• 39.8% 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: 96.5% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma y (- (log z) t) (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(fma(y, (log(z) - t), (a * (log1p(-z) - b))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(fma(y, Float64(log(z) - t), Float64(a * Float64(log1p(Float64(-z)) - b)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

   1. fma-define 97.4%

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

   2. sub-neg 97.4%

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

   3. log1p-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-36} \lor \neg \left(y \leq 1.12 \cdot 10^{-127}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(\left(-b\right) - z\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5e-36) (not (<= y 1.12e-127)))
   (* x (exp (* y (- (log z) t))))
   (* x (pow E (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-36) || !(y <= 1.12e-127)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * pow(((double) M_E), (a * (-b - z)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-36) || !(y <= 1.12e-127)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.pow(Math.E, (a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5e-36) or not (y <= 1.12e-127):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.pow(math.e, (a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5e-36) || !(y <= 1.12e-127))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * (exp(1) ^ Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5e-36) || ~((y <= 1.12e-127)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * (2.71828182845904523536 ^ (a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5e-36], N[Not[LessEqual[y, 1.12e-127]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[E, N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000004e-36 or 1.1199999999999999e-127 < y

   1. Initial program 98.7%

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

      1. fma-define 99.3%

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

      2. sub-neg 99.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 86.5%

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

   if -5.00000000000000004e-36 < y < 1.1199999999999999e-127

   1. Initial program 94.5%

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

      1. fma-define 94.5%

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

      2. sub-neg 94.5%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 90.2%

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

      1. *-un-lft-identity 90.2%

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

      2. exp-prod 90.2%

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

      3. sub-neg 90.2%

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

      4. log1p-define 95.7%

         \[\leadsto x \cdot {\left(e^{1}\right)}^{\left(a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)\right)} \]

   7. Applied egg-rr 95.7%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   8. Step-by-step derivation

      1. exp-1-e 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in z around 0 95.7%

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

       1. associate-*r* 95.7%

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

       2. associate-*r* 95.7%

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

       3. distribute-lft-out 95.7%

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

       4. mul-1-neg 95.7%

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

   12. Simplified 95.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-36} \lor \neg \left(y \leq 1.12 \cdot 10^{-127}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(\left(-b\right) - z\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * (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[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in z around 0 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

   3. distribute-lft-out 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 74.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.3 \cdot 10^{-76} \lor \neg \left(a \leq 7 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -8.3e-76) (not (<= a 7e-38)))
   (* x (exp (* a (- (- b) z))))
   (* x (exp (* y (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.3e-76) || !(a <= 7e-38)) {
		tmp = x * exp((a * (-b - z)));
	} else {
		tmp = x * exp((y * -t));
	}
	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) :: tmp
    if ((a <= (-8.3d-76)) .or. (.not. (a <= 7d-38))) then
        tmp = x * exp((a * (-b - z)))
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.3e-76) || !(a <= 7e-38)) {
		tmp = x * Math.exp((a * (-b - z)));
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -8.3e-76) or not (a <= 7e-38):
		tmp = x * math.exp((a * (-b - z)))
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -8.3e-76) || !(a <= 7e-38))
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -8.3e-76) || ~((a <= 7e-38)))
		tmp = x * exp((a * (-b - z)));
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -8.3e-76], N[Not[LessEqual[a, 7e-38]], $MachinePrecision]], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.3000000000000002e-76 or 7.0000000000000003e-38 < a

   1. Initial program 94.8%

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

      1. fma-define 95.4%

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

      2. sub-neg 95.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.4%

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

   6. Taylor expanded in z around 0 83.9%

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

      1. associate-*r* 99.3%

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

      2. associate-*r* 99.3%

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

      3. distribute-lft-out 99.3%

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

      4. mul-1-neg 99.3%

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

   8. Simplified 83.9%

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

   if -8.3000000000000002e-76 < a < 7.0000000000000003e-38

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - z\right) - b} \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right) \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right)}} \]

      2. pow3 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\log \left(1 - z\right) - b}\right)}^{3}}} \]

      3. sub-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-z\right)\right)} - b}\right)}^{3}} \]

      4. log1p-undefine 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-z\right)} - b}\right)}^{3}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right) - b}\right)}^{3}} \]

      6. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) - b}\right)}^{3}} \]

      7. sqr-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot z}}\right) - b}\right)}^{3}} \]

      8. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) - b}\right)}^{3}} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{z}\right) - b}\right)}^{3}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(z\right) - b}\right)}^{3}}} \]

   5. Taylor expanded in t around inf 82.1%

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

      1. associate-*r* 82.1%

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

      2. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.3 \cdot 10^{-76} \lor \neg \left(a \leq 7 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(\left(-b\right) - z\right)\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot {e}^{t\_1}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (- (- b) z))))
   (if (<= a -5.4e-72)
     (* x (pow E t_1))
     (if (<= a 1.8e-35) (* x (exp (* y (- t)))) (* x (exp t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (-b - z);
	double tmp;
	if (a <= -5.4e-72) {
		tmp = x * pow(((double) M_E), t_1);
	} else if (a <= 1.8e-35) {
		tmp = x * exp((y * -t));
	} else {
		tmp = x * exp(t_1);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (-b - z);
	double tmp;
	if (a <= -5.4e-72) {
		tmp = x * Math.pow(Math.E, t_1);
	} else if (a <= 1.8e-35) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.exp(t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (-b - z)
	tmp = 0
	if a <= -5.4e-72:
		tmp = x * math.pow(math.e, t_1)
	elif a <= 1.8e-35:
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.exp(t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(Float64(-b) - z))
	tmp = 0.0
	if (a <= -5.4e-72)
		tmp = Float64(x * (exp(1) ^ t_1));
	elseif (a <= 1.8e-35)
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = Float64(x * exp(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (-b - z);
	tmp = 0.0;
	if (a <= -5.4e-72)
		tmp = x * (2.71828182845904523536 ^ t_1);
	elseif (a <= 1.8e-35)
		tmp = x * exp((y * -t));
	else
		tmp = x * exp(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.4e-72], N[(x * N[Power[E, t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e-35], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.4e-72

   1. Initial program 94.5%

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

      1. fma-define 94.5%

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

      2. sub-neg 94.5%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. exp-prod 74.0%

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

      3. sub-neg 74.0%

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

      4. log1p-define 84.8%

         \[\leadsto x \cdot {\left(e^{1}\right)}^{\left(a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)\right)} \]

   7. Applied egg-rr 84.8%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   8. Step-by-step derivation

      1. exp-1-e 84.8%

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

   9. Simplified 84.8%

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

   10. Taylor expanded in z around 0 84.8%

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

       1. associate-*r* 84.8%

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

       2. associate-*r* 84.8%

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

       3. distribute-lft-out 84.8%

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

       4. mul-1-neg 84.8%

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

   12. Simplified 84.8%

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

   if -5.4e-72 < a < 1.80000000000000009e-35

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - z\right) - b} \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right) \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right)}} \]

      2. pow3 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\log \left(1 - z\right) - b}\right)}^{3}}} \]

      3. sub-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-z\right)\right)} - b}\right)}^{3}} \]

      4. log1p-undefine 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-z\right)} - b}\right)}^{3}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right) - b}\right)}^{3}} \]

      6. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) - b}\right)}^{3}} \]

      7. sqr-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot z}}\right) - b}\right)}^{3}} \]

      8. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) - b}\right)}^{3}} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{z}\right) - b}\right)}^{3}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(z\right) - b}\right)}^{3}}} \]

   5. Taylor expanded in t around inf 82.1%

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

      1. associate-*r* 82.1%

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

      2. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

   if 1.80000000000000009e-35 < a

   1. Initial program 95.0%

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

      1. fma-define 96.2%

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

      2. sub-neg 96.2%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 80.5%

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

   6. Taylor expanded in z around 0 83.0%

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

      1. associate-*r* 98.7%

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

      2. associate-*r* 98.7%

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

      3. distribute-lft-out 98.7%

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

      4. mul-1-neg 98.7%

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

   8. Simplified 83.0%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(b + z\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(\left(-b\right) - z\right)\right)}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-62} \lor \neg \left(a \leq 2.1 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -6.6e-62) (not (<= a 2.1e-35)))
   (* x (exp (* a (- b))))
   (* x (exp (* y (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -6.6e-62) || !(a <= 2.1e-35)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp((y * -t));
	}
	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) :: tmp
    if ((a <= (-6.6d-62)) .or. (.not. (a <= 2.1d-35))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -6.6e-62) || !(a <= 2.1e-35)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -6.6e-62) or not (a <= 2.1e-35):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -6.6e-62) || !(a <= 2.1e-35))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -6.6e-62) || ~((a <= 2.1e-35)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -6.6e-62], N[Not[LessEqual[a, 2.1e-35]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.60000000000000009e-62 or 2.1e-35 < a

   1. Initial program 94.8%

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

      1. fma-define 95.4%

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

      2. sub-neg 95.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.4%

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

   6. Taylor expanded in z around 0 76.8%

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

      1. mul-1-neg 76.8%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 76.8%

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

   8. Simplified 76.8%

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

   if -6.60000000000000009e-62 < a < 2.1e-35

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - z\right) - b} \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right) \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right)}} \]

      2. pow3 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\log \left(1 - z\right) - b}\right)}^{3}}} \]

      3. sub-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-z\right)\right)} - b}\right)}^{3}} \]

      4. log1p-undefine 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-z\right)} - b}\right)}^{3}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right) - b}\right)}^{3}} \]

      6. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) - b}\right)}^{3}} \]

      7. sqr-neg 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot z}}\right) - b}\right)}^{3}} \]

      8. sqrt-unprod 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) - b}\right)}^{3}} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{z}\right) - b}\right)}^{3}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(z\right) - b}\right)}^{3}}} \]

   5. Taylor expanded in t around inf 82.1%

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

      1. associate-*r* 82.1%

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

      2. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-62} \lor \neg \left(a \leq 2.1 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.14 \cdot 10^{+64} \lor \neg \left(y \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.14e+64) (not (<= y 2e+19)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.14e+64) || !(y <= 2e+19)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	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) :: tmp
    if ((y <= (-1.14d+64)) .or. (.not. (y <= 2d+19))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.14e+64) || !(y <= 2e+19)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.14e+64) or not (y <= 2e+19):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.14e+64) || !(y <= 2e+19))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.14e+64) || ~((y <= 2e+19)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.14e+64], N[Not[LessEqual[y, 2e+19]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14e64 or 2e19 < y

   1. Initial program 99.0%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 92.1%

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

   6. Taylor expanded in t around 0 71.5%

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

   if -1.14e64 < y < 2e19

   1. Initial program 95.7%

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

      1. fma-define 95.7%

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

      2. sub-neg 95.7%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.6%

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

   6. Taylor expanded in z around 0 77.0%

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

      1. mul-1-neg 77.0%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 77.0%

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

   8. Simplified 77.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.14 \cdot 10^{+64} \lor \neg \left(y \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.14 \cdot 10^{+64} \lor \neg \left(y \leq 3 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{z \cdot \left(-a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.14e+64) (not (<= y 3e-7)))
   (* x (pow z y))
   (* x (exp (* z (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.14e+64) || !(y <= 3e-7)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((z * -a));
	}
	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) :: tmp
    if ((y <= (-1.14d+64)) .or. (.not. (y <= 3d-7))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((z * -a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.14e+64) || !(y <= 3e-7)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((z * -a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.14e+64) or not (y <= 3e-7):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((z * -a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.14e+64) || !(y <= 3e-7))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(z * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.14e+64) || ~((y <= 3e-7)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((z * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.14e+64], N[Not[LessEqual[y, 3e-7]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(z * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14e64 or 2.9999999999999999e-7 < y

   1. Initial program 99.1%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 91.6%

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

   6. Taylor expanded in t around 0 69.4%

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

   if -1.14e64 < y < 2.9999999999999999e-7

   1. Initial program 95.5%

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

      1. fma-define 95.5%

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

      2. sub-neg 95.5%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. exp-prod 78.4%

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

      3. sub-neg 78.4%

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

      4. log1p-define 82.9%

         \[\leadsto x \cdot {\left(e^{1}\right)}^{\left(a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)\right)} \]

   7. Applied egg-rr 82.9%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   8. Step-by-step derivation

      1. exp-1-e 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in z around 0 82.9%

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

       1. associate-*r* 82.9%

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

       2. associate-*r* 82.9%

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

       3. distribute-lft-out 82.9%

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

       4. mul-1-neg 82.9%

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

   12. Simplified 82.9%

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

   13. Taylor expanded in b around 0 54.9%

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

       1. exp-prod 54.9%

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

       2. log-E 54.9%

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

       3. associate-*r* 54.9%

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

       4. *-rgt-identity 54.9%

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

       5. exp-prod 54.9%

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

       6. mul-1-neg 54.9%

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

   15. Simplified 54.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.14 \cdot 10^{+64} \lor \neg \left(y \leq 3 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{z \cdot \left(-a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.3e+16) (* x (- 1.0 (* y t))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.3e+16) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * pow(z, y);
	}
	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) :: tmp
    if (t <= (-2.3d+16)) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.3e+16) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.3e+16:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.3e+16)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.3e+16)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.3e+16], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3e16

   1. Initial program 98.5%

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

      1. add-cube-cbrt 98.5%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - z\right) - b} \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right) \cdot \sqrt[3]{\log \left(1 - z\right) - b}\right)}} \]

      2. pow3 98.5%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\log \left(1 - z\right) - b}\right)}^{3}}} \]

      3. sub-neg 98.5%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-z\right)\right)} - b}\right)}^{3}} \]

      4. log1p-undefine 99.8%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-z\right)} - b}\right)}^{3}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right) - b}\right)}^{3}} \]

      6. sqrt-unprod 96.9%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) - b}\right)}^{3}} \]

      7. sqr-neg 96.9%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot z}}\right) - b}\right)}^{3}} \]

      8. sqrt-unprod 96.9%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) - b}\right)}^{3}} \]

      9. add-sqr-sqrt 96.9%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{z}\right) - b}\right)}^{3}} \]

   4. Applied egg-rr 96.9%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(z\right) - b}\right)}^{3}}} \]

   5. Taylor expanded in t around inf 80.7%

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

      1. associate-*r* 80.7%

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

      2. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in t around 0 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

      3. unsub-neg 31.6%

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

   10. Simplified 31.6%

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

   if -2.3e16 < t

   1. Initial program 96.5%

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

      1. fma-define 97.0%

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

      2. sub-neg 97.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 71.0%

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

   6. Taylor expanded in t around 0 63.3%

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

Alternative 10: 33.7% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-30}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 72000000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.6e-30)
   (* b (- (/ x b) (* x a)))
   (if (<= y 72000000000.0) (* x (- 1.0 (* a (+ z b)))) (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.6e-30) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 72000000000.0) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = x * (a * -b);
	}
	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) :: tmp
    if (y <= (-2.6d-30)) then
        tmp = b * ((x / b) - (x * a))
    else if (y <= 72000000000.0d0) then
        tmp = x * (1.0d0 - (a * (z + b)))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.6e-30) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 72000000000.0) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.6e-30:
		tmp = b * ((x / b) - (x * a))
	elif y <= 72000000000.0:
		tmp = x * (1.0 - (a * (z + b)))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.6e-30)
		tmp = Float64(b * Float64(Float64(x / b) - Float64(x * a)));
	elseif (y <= 72000000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.6e-30)
		tmp = b * ((x / b) - (x * a));
	elseif (y <= 72000000000.0)
		tmp = x * (1.0 - (a * (z + b)));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.6e-30], N[(b * N[(N[(x / b), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 72000000000.0], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999987e-30

   1. Initial program 96.7%

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

      1. fma-define 98.4%

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

      2. sub-neg 98.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 39.8%

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

   6. Taylor expanded in z around 0 39.8%

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

      1. mul-1-neg 39.8%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 39.8%

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

   8. Simplified 39.8%

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

   9. Taylor expanded in a around 0 15.9%

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

       1. neg-mul-1 15.9%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 15.9%

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

   11. Simplified 15.9%

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

   12. Taylor expanded in b around inf 26.5%

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

       1. +-commutative 26.5%

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

       2. mul-1-neg 26.5%

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

       3. sub-neg 26.5%

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

       4. *-commutative 26.5%

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

   14. Simplified 26.5%

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

   if -2.59999999999999987e-30 < y < 7.2e10

   1. Initial program 95.9%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 79.8%

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

      1. *-un-lft-identity 79.8%

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

      2. exp-prod 79.8%

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

      3. sub-neg 79.8%

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

      4. log1p-define 83.9%

         \[\leadsto x \cdot {\left(e^{1}\right)}^{\left(a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)\right)} \]

   7. Applied egg-rr 83.9%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   8. Step-by-step derivation

      1. exp-1-e 83.9%

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

   9. Simplified 83.9%

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

   10. Taylor expanded in z around 0 83.9%

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

       1. associate-*r* 83.9%

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

       2. associate-*r* 83.9%

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

       3. distribute-lft-out 83.9%

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

       4. mul-1-neg 83.9%

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

   12. Simplified 83.9%

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

   13. Taylor expanded in a around 0 48.7%

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

       1. mul-1-neg 48.7%

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

       2. unsub-neg 48.7%

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

       3. log-E 48.7%

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

       4. *-lft-identity 48.7%

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

   15. Simplified 48.7%

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

   if 7.2e10 < y

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 43.3%

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

   6. Taylor expanded in z around 0 43.3%

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

      1. mul-1-neg 43.3%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in a around 0 12.1%

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

       1. neg-mul-1 12.1%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 12.1%

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

   11. Simplified 12.1%

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

   12. Taylor expanded in a around inf 28.1%

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

       1. mul-1-neg 28.1%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

       4. distribute-rgt-neg-in 31.4%

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

       5. distribute-rgt-neg-in 31.4%

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

   14. Simplified 31.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-30}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 72000000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 6800000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e-35)
   (* b (- (/ x b) (* x a)))
   (if (<= y 6800000000.0) (* x (- 1.0 (* a b))) (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-35) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 6800000000.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	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) :: tmp
    if (y <= (-1.6d-35)) then
        tmp = b * ((x / b) - (x * a))
    else if (y <= 6800000000.0d0) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-35) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 6800000000.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.6e-35:
		tmp = b * ((x / b) - (x * a))
	elif y <= 6800000000.0:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e-35)
		tmp = Float64(b * Float64(Float64(x / b) - Float64(x * a)));
	elseif (y <= 6800000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.6e-35)
		tmp = b * ((x / b) - (x * a));
	elseif (y <= 6800000000.0)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e-35], N[(b * N[(N[(x / b), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6800000000.0], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5999999999999999e-35

   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. Step-by-step derivation

      1. fma-define 98.4%

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

      2. sub-neg 98.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 39.2%

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

   6. Taylor expanded in z around 0 39.2%

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

      1. mul-1-neg 39.2%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in a around 0 15.7%

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

       1. neg-mul-1 15.7%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 15.7%

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

   11. Simplified 15.7%

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

   12. Taylor expanded in b around inf 26.2%

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

       1. +-commutative 26.2%

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

       2. mul-1-neg 26.2%

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

       3. sub-neg 26.2%

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

       4. *-commutative 26.2%

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

   14. Simplified 26.2%

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

   if -1.5999999999999999e-35 < y < 6.8e9

   1. Initial program 95.8%

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

      1. fma-define 95.8%

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

      2. sub-neg 95.8%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 80.3%

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

   6. Taylor expanded in z around 0 79.6%

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

      1. mul-1-neg 79.6%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in a around 0 48.3%

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

       1. neg-mul-1 48.3%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 48.3%

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

   11. Simplified 48.3%

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

   if 6.8e9 < y

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 43.3%

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

   6. Taylor expanded in z around 0 43.3%

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

      1. mul-1-neg 43.3%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in a around 0 12.1%

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

       1. neg-mul-1 12.1%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 12.1%

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

   11. Simplified 12.1%

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

   12. Taylor expanded in a around inf 28.1%

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

       1. mul-1-neg 28.1%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

       4. distribute-rgt-neg-in 31.4%

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

       5. distribute-rgt-neg-in 31.4%

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

   14. Simplified 31.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 32.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 8600000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.5e-36)
   (* a (* x (- b)))
   (if (<= y 8600000000.0) (* x (- 1.0 (* a b))) (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e-36) {
		tmp = a * (x * -b);
	} else if (y <= 8600000000.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	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) :: tmp
    if (y <= (-8.5d-36)) then
        tmp = a * (x * -b)
    else if (y <= 8600000000.0d0) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e-36) {
		tmp = a * (x * -b);
	} else if (y <= 8600000000.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.5e-36:
		tmp = a * (x * -b)
	elif y <= 8600000000.0:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.5e-36)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (y <= 8600000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.5e-36)
		tmp = a * (x * -b);
	elseif (y <= 8600000000.0)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.5e-36], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8600000000.0], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000007e-36

   1. Initial program 96.9%

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

      1. fma-define 98.4%

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

      2. sub-neg 98.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 38.7%

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

   6. Taylor expanded in z around 0 38.7%

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

      1. mul-1-neg 38.7%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 38.7%

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

   8. Simplified 38.7%

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

   9. Taylor expanded in a around 0 15.6%

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

       1. neg-mul-1 15.6%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 15.6%

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

   11. Simplified 15.6%

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

   12. Taylor expanded in a around inf 24.2%

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

   if -8.5000000000000007e-36 < y < 8.6e9

   1. Initial program 95.8%

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

      1. fma-define 95.8%

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

      2. sub-neg 95.8%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 80.8%

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

   6. Taylor expanded in z around 0 80.1%

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

      1. mul-1-neg 80.1%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in a around 0 48.6%

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

       1. neg-mul-1 48.6%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 48.6%

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

   11. Simplified 48.6%

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

   if 8.6e9 < y

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 43.3%

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

   6. Taylor expanded in z around 0 43.3%

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

      1. mul-1-neg 43.3%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in a around 0 12.1%

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

       1. neg-mul-1 12.1%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 12.1%

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

   11. Simplified 12.1%

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

   12. Taylor expanded in a around inf 28.1%

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

       1. mul-1-neg 28.1%

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

       2. associate-*r* 31.4%

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

       3. *-commutative 31.4%

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

       4. distribute-rgt-neg-in 31.4%

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

       5. distribute-rgt-neg-in 31.4%

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

   14. Simplified 31.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 8600000000:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36} \lor \neg \left(y \leq 1.15 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.5e-36) (not (<= y 1.15e+16))) (* x (* a (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.5e-36) || !(y <= 1.15e+16)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	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) :: tmp
    if ((y <= (-8.5d-36)) .or. (.not. (y <= 1.15d+16))) then
        tmp = x * (a * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.5e-36) || !(y <= 1.15e+16)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.5e-36) or not (y <= 1.15e+16):
		tmp = x * (a * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.5e-36) || !(y <= 1.15e+16))
		tmp = Float64(x * Float64(a * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -8.5e-36) || ~((y <= 1.15e+16)))
		tmp = x * (a * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.5e-36], N[Not[LessEqual[y, 1.15e+16]], $MachinePrecision]], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000007e-36 or 1.15e16 < y

   1. Initial program 98.3%

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

      1. fma-define 99.2%

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

      2. sub-neg 99.2%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 39.8%

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

   6. Taylor expanded in z around 0 39.8%

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

      1. mul-1-neg 39.8%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 39.8%

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

   8. Simplified 39.8%

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

   9. Taylor expanded in a around 0 14.1%

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

       1. neg-mul-1 14.1%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in a around inf 26.4%

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

       1. mul-1-neg 26.4%

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

       2. associate-*r* 27.2%

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

       3. *-commutative 27.2%

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

       4. distribute-rgt-neg-in 27.2%

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

       5. distribute-rgt-neg-in 27.2%

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

   14. Simplified 27.2%

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

   if -8.5000000000000007e-36 < y < 1.15e16

   1. Initial program 95.9%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 81.1%

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

   6. Taylor expanded in a around 0 42.0%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36} \lor \neg \left(y \leq 1.15 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-36}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6e-36) (* a (* x (- b))) (if (<= y 1.15e+16) x (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6e-36) {
		tmp = a * (x * -b);
	} else if (y <= 1.15e+16) {
		tmp = x;
	} else {
		tmp = x * (a * -b);
	}
	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) :: tmp
    if (y <= (-6d-36)) then
        tmp = a * (x * -b)
    else if (y <= 1.15d+16) then
        tmp = x
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6e-36) {
		tmp = a * (x * -b);
	} else if (y <= 1.15e+16) {
		tmp = x;
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6e-36:
		tmp = a * (x * -b)
	elif y <= 1.15e+16:
		tmp = x
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6e-36)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (y <= 1.15e+16)
		tmp = x;
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6e-36)
		tmp = a * (x * -b);
	elseif (y <= 1.15e+16)
		tmp = x;
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6e-36], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+16], x, N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000003e-36

   1. Initial program 96.9%

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

      1. fma-define 98.4%

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

      2. sub-neg 98.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 38.7%

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

   6. Taylor expanded in z around 0 38.7%

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

      1. mul-1-neg 38.7%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 38.7%

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

   8. Simplified 38.7%

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

   9. Taylor expanded in a around 0 15.6%

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

       1. neg-mul-1 15.6%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 15.6%

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

   11. Simplified 15.6%

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

   12. Taylor expanded in a around inf 24.2%

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

   if -6.0000000000000003e-36 < y < 1.15e16

   1. Initial program 95.9%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 81.1%

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

   6. Taylor expanded in a around 0 42.0%

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

   if 1.15e16 < y

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 41.2%

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

   6. Taylor expanded in z around 0 41.2%

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

      1. mul-1-neg 41.2%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-rgt-neg-out 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in a around 0 12.4%

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

       1. neg-mul-1 12.4%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

       2. unsub-neg 12.4%

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

   11. Simplified 12.4%

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

   12. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. associate-*r* 32.4%

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

       3. *-commutative 32.4%

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

       4. distribute-rgt-neg-in 32.4%

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

       5. distribute-rgt-neg-in 32.4%

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

   14. Simplified 32.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-36}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 97.0%

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

   1. fma-define 97.4%

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

   2. sub-neg 97.4%

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

   3. log1p-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 62.3%

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

6. Taylor expanded in a around 0 24.7%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))