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

Percentage Accurate: 96.8% → 99.7%

Time: 16.7s

Alternatives: 13

Speedup: 1.5×

• Could not converge on a ground truth

  1. x = 1.3618056079162772e-82
  2. y = -2.2846395409128378e-265
  3. z = 7.260430565363525e-91
  4. t = 8.296439917658668e+116
  5. a = -3.133740969661257e-192
  6. b = -7.676070296441502e-222

• 41.4% 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: 96.8% 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 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 98.8%

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

3. Simplified 98.8%

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

Alternative 2: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-9} \lor \neg \left(y \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.35e-9) (not (<= y 8.5e-9)))
   (* x (exp (* y (- (log z) t))))
   (* x (pow E (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.35e-9) || !(y <= 8.5e-9)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * pow(((double) M_E), (a * -b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.35e-9) || !(y <= 8.5e-9)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.pow(Math.E, (a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.35e-9) or not (y <= 8.5e-9):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.pow(math.e, (a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.35e-9) || !(y <= 8.5e-9))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * (exp(1) ^ Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.35e-9) || ~((y <= 8.5e-9)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * (2.71828182845904523536 ^ (a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.35e-9], N[Not[LessEqual[y, 8.5e-9]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[E, N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001e-9 or 8.5e-9 < y

   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 96.9%

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

      2. sub-neg 96.9%

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

      3. log1p-define 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in a around 0 85.4%

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

   if -1.3500000000000001e-9 < y < 8.5e-9

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. exp-prod 100.0%

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

      3. fma-define 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. fmm-undef 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in b around inf 83.9%

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

      1. neg-mul-1 83.9%

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

      2. distribute-rgt-neg-in 83.9%

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

   10. Simplified 83.9%

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

       1. *-un-lft-identity 83.9%

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

       2. exp-1-e 83.9%

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

   12. Applied egg-rr 83.9%

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

       1. *-lft-identity 83.9%

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

   14. Simplified 83.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-9} \lor \neg \left(y \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% 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 95.7%

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

3. Taylor expanded in z around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-*r* 98.8%

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

   3. associate-*r* 98.8%

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

   4. distribute-lft-out 98.8%

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

   5. mul-1-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 70.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+67} \lor \neg \left(t \leq 3.5 \cdot 10^{+180}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3.8e+67) (not (<= t 3.5e+180)))
   (* x (exp (* y (- t))))
   (* x (pow E (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.8e+67) || !(t <= 3.5e+180)) {
		tmp = x * exp((y * -t));
	} else {
		tmp = x * pow(((double) M_E), (a * -b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.8e+67) || !(t <= 3.5e+180)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.pow(Math.E, (a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -3.8e+67) or not (t <= 3.5e+180):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.pow(math.e, (a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.8e+67) || !(t <= 3.5e+180))
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = Float64(x * (exp(1) ^ Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -3.8e+67) || ~((t <= 3.5e+180)))
		tmp = x * exp((y * -t));
	else
		tmp = x * (2.71828182845904523536 ^ (a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3.8e+67], N[Not[LessEqual[t, 3.5e+180]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[E, N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8000000000000002e67 or 3.4999999999999998e180 < t

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. associate-*r* 96.4%

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

      3. associate-*r* 96.4%

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

      4. distribute-lft-out 96.4%

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

      5. mul-1-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf 87.1%

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

      1. mul-1-neg 87.1%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. *-commutative 87.1%

         \[\leadsto x \cdot e^{-\color{blue}{y \cdot t}} \]

      3. distribute-lft-neg-in 87.1%

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

   8. Simplified 87.1%

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

   if -3.8000000000000002e67 < t < 3.4999999999999998e180

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. exp-prod 99.9%

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

      3. fma-define 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. fmm-undef 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in b around inf 70.0%

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

      1. neg-mul-1 70.0%

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

      2. distribute-rgt-neg-in 70.0%

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

   10. Simplified 70.0%

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

       1. *-un-lft-identity 70.0%

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

       2. exp-1-e 70.0%

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

   12. Applied egg-rr 70.0%

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

       1. *-lft-identity 70.0%

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

   14. Simplified 70.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+67} \lor \neg \left(t \leq 3.5 \cdot 10^{+180}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+67} \lor \neg \left(t \leq 5.5 \cdot 10^{+181}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -4.4e+67) (not (<= t 5.5e+181)))
   (* x (exp (* y (- t))))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.4e+67) || !(t <= 5.5e+181)) {
		tmp = x * exp((y * -t));
	} 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 ((t <= (-4.4d+67)) .or. (.not. (t <= 5.5d+181))) then
        tmp = x * exp((y * -t))
    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 ((t <= -4.4e+67) || !(t <= 5.5e+181)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -4.4e+67) or not (t <= 5.5e+181):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -4.4e+67) || !(t <= 5.5e+181))
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	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 ((t <= -4.4e+67) || ~((t <= 5.5e+181)))
		tmp = x * exp((y * -t));
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -4.4e+67], N[Not[LessEqual[t, 5.5e+181]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.4e67 or 5.49999999999999991e181 < t

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. associate-*r* 96.4%

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

      3. associate-*r* 96.4%

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

      4. distribute-lft-out 96.4%

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

      5. mul-1-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf 87.1%

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

      1. mul-1-neg 87.1%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. *-commutative 87.1%

         \[\leadsto x \cdot e^{-\color{blue}{y \cdot t}} \]

      3. distribute-lft-neg-in 87.1%

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

   8. Simplified 87.1%

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

   if -4.4e67 < t < 5.49999999999999991e181

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 70.0%

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

      1. neg-mul-1 70.0%

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

      2. distribute-lft-neg-in 70.0%

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

      3. *-commutative 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+67} \lor \neg \left(t \leq 5.5 \cdot 10^{+181}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+103} \lor \neg \left(y \leq 2.5 \cdot 10^{+221}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5e+103) (not (<= y 2.5e+221)))
   (* 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 <= -5e+103) || !(y <= 2.5e+221)) {
		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 <= (-5d+103)) .or. (.not. (y <= 2.5d+221))) 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 <= -5e+103) || !(y <= 2.5e+221)) {
		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 <= -5e+103) or not (y <= 2.5e+221):
		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 <= -5e+103) || !(y <= 2.5e+221))
		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 <= -5e+103) || ~((y <= 2.5e+221)))
		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, -5e+103], N[Not[LessEqual[y, 2.5e+221]], $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 < -5e103 or 2.5000000000000001e221 < y

   1. Initial program 96.0%

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

      1. fma-define 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-define 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in a around 0 98.0%

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

   6. Taylor expanded in t around 0 76.4%

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

   if -5e103 < y < 2.5000000000000001e221

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. associate-*r* 99.4%

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

      4. distribute-lft-out 99.4%

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

      5. mul-1-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in b around inf 72.0%

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

      1. neg-mul-1 72.0%

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

      2. distribute-lft-neg-in 72.0%

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

      3. *-commutative 72.0%

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

   8. Simplified 72.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+103} \lor \neg \left(y \leq 2.5 \cdot 10^{+221}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+285}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;t \leq -125000000000:\\ \;\;\;\;x \cdot e^{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.02e+285)
   (* x (- 1.0 (* y t)))
   (if (<= t -125000000000.0) (* x (exp (* a b))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.02e+285) {
		tmp = x * (1.0 - (y * t));
	} else if (t <= -125000000000.0) {
		tmp = x * exp((a * b));
	} 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 <= (-1.02d+285)) then
        tmp = x * (1.0d0 - (y * t))
    else if (t <= (-125000000000.0d0)) then
        tmp = x * exp((a * b))
    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 <= -1.02e+285) {
		tmp = x * (1.0 - (y * t));
	} else if (t <= -125000000000.0) {
		tmp = x * Math.exp((a * b));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.02e+285:
		tmp = x * (1.0 - (y * t))
	elif t <= -125000000000.0:
		tmp = x * math.exp((a * b))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.02e+285)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (t <= -125000000000.0)
		tmp = Float64(x * exp(Float64(a * b)));
	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 <= -1.02e+285)
		tmp = x * (1.0 - (y * t));
	elseif (t <= -125000000000.0)
		tmp = x * exp((a * b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.02e+285], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -125000000000.0], N[(x * N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.01999999999999996e285

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. associate-*r* 88.9%

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

      3. associate-*r* 88.9%

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

      4. distribute-lft-out 88.9%

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

      5. mul-1-neg 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. mul-1-neg 100.0%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. *-commutative 100.0%

         \[\leadsto x \cdot e^{-\color{blue}{y \cdot t}} \]

      3. distribute-lft-neg-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0 78.2%

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

       1. mul-1-neg 78.2%

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

       2. *-commutative 78.2%

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

       3. unsub-neg 78.2%

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

   11. Simplified 78.2%

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

   if -1.01999999999999996e285 < t < -1.25e11

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-*r* 98.4%

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

      3. associate-*r* 98.4%

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

      4. distribute-lft-out 98.4%

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

      5. mul-1-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 56.4%

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

      1. neg-mul-1 56.4%

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

      2. distribute-lft-neg-in 56.4%

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

      3. *-commutative 56.4%

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

   8. Simplified 56.4%

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

      1. neg-sub0 56.4%

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 - a\right)}} \]

      2. sub-neg 56.4%

         \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 + \left(-a\right)\right)}} \]

      3. add-sqr-sqrt 25.2%

         \[\leadsto x \cdot e^{b \cdot \left(0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)} \]

      4. sqrt-unprod 41.0%

         \[\leadsto x \cdot e^{b \cdot \left(0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)} \]

      5. sqr-neg 41.0%

         \[\leadsto x \cdot e^{b \cdot \left(0 + \sqrt{\color{blue}{a \cdot a}}\right)} \]

      6. sqrt-unprod 15.8%

         \[\leadsto x \cdot e^{b \cdot \left(0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)} \]

      7. add-sqr-sqrt 36.4%

         \[\leadsto x \cdot e^{b \cdot \left(0 + \color{blue}{a}\right)} \]

   10. Applied egg-rr 36.4%

       \[\leadsto x \cdot e^{b \cdot \color{blue}{\left(0 + a\right)}} \]
   11. Step-by-step derivation

       1. +-lft-identity 36.4%

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

   12. Simplified 36.4%

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

   if -1.25e11 < t

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 64.0%

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

   6. Taylor expanded in t around 0 58.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+285}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;t \leq -125000000000:\\ \;\;\;\;x \cdot e^{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1400:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1400.0) (* 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 <= -1400.0) {
		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 <= (-1400.0d0)) 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 <= -1400.0) {
		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 <= -1400.0:
		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 <= -1400.0)
		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 <= -1400.0)
		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, -1400.0], 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 < -1400

   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. Add Preprocessing

   3. Taylor expanded in z around 0 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-*r* 97.4%

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

      3. associate-*r* 97.4%

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

      4. distribute-lft-out 97.4%

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

      5. mul-1-neg 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in t around inf 77.0%

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

      1. mul-1-neg 77.0%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. *-commutative 77.0%

         \[\leadsto x \cdot e^{-\color{blue}{y \cdot t}} \]

      3. distribute-lft-neg-in 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in y around 0 33.4%

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

       1. mul-1-neg 33.4%

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

       2. *-commutative 33.4%

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

       3. unsub-neg 33.4%

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

   11. Simplified 33.4%

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

   if -1400 < t

   1. Initial program 96.1%

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

      1. fma-define 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-define 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around 0 64.5%

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

   6. Taylor expanded in t around 0 59.3%

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

Alternative 9: 32.9% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+14} \lor \neg \left(y \leq 0.1\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.3e+14) (not (<= y 0.1)))
   (* a (* x (- b)))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.3e+14) || !(y <= 0.1)) {
		tmp = a * (x * -b);
	} else {
		tmp = x * (1.0 - (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 <= (-1.3d+14)) .or. (.not. (y <= 0.1d0))) then
        tmp = a * (x * -b)
    else
        tmp = x * (1.0d0 - (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 <= -1.3e+14) || !(y <= 0.1)) {
		tmp = a * (x * -b);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.3e+14) or not (y <= 0.1):
		tmp = a * (x * -b)
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.3e+14) || !(y <= 0.1))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.3e+14) || ~((y <= 0.1)))
		tmp = a * (x * -b);
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.3e+14], N[Not[LessEqual[y, 0.1]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e14 or 0.10000000000000001 < y

   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. Add Preprocessing

   3. Taylor expanded in z around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-*r* 97.5%

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

      3. associate-*r* 97.5%

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

      4. distribute-lft-out 97.5%

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

      5. mul-1-neg 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in b around inf 38.8%

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

      1. neg-mul-1 38.8%

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

      2. distribute-lft-neg-in 38.8%

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

      3. *-commutative 38.8%

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

   8. Simplified 38.8%

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

   9. Taylor expanded in b around 0 10.2%

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

       1. neg-mul-1 10.2%

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

       2. unsub-neg 10.2%

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

   11. Simplified 10.2%

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

   12. Taylor expanded in a around inf 21.4%

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

       1. mul-1-neg 21.4%

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

       2. *-commutative 21.4%

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

       3. distribute-rgt-neg-in 21.4%

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

       4. distribute-rgt-neg-in 21.4%

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

   14. Simplified 21.4%

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

   if -1.3e14 < y < 0.10000000000000001

   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. Add Preprocessing

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 81.5%

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

      1. neg-mul-1 81.5%

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

      2. distribute-lft-neg-in 81.5%

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

      3. *-commutative 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in b around 0 44.6%

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

       1. neg-mul-1 44.6%

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

       2. unsub-neg 44.6%

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

   11. Simplified 44.6%

       \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+14} \lor \neg \left(y \leq 0.1\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+185}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.6e+24)
   (* x (* a (- b)))
   (if (<= a 3.2e+185) (* x (- 1.0 (* y t))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.6e+24) {
		tmp = x * (a * -b);
	} else if (a <= 3.2e+185) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * (x * -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 (a <= (-6.6d+24)) then
        tmp = x * (a * -b)
    else if (a <= 3.2d+185) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = a * (x * -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 (a <= -6.6e+24) {
		tmp = x * (a * -b);
	} else if (a <= 3.2e+185) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6.6e+24:
		tmp = x * (a * -b)
	elif a <= 3.2e+185:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6.6e+24)
		tmp = Float64(x * Float64(a * Float64(-b)));
	elseif (a <= 3.2e+185)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6.6e+24)
		tmp = x * (a * -b);
	elseif (a <= 3.2e+185)
		tmp = x * (1.0 - (y * t));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6.6e+24], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.2e+185], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.5999999999999998e24

   1. Initial program 84.3%

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

   3. Taylor expanded in z around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-*r* 98.2%

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

      3. associate-*r* 98.2%

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

      4. distribute-lft-out 98.2%

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

      5. mul-1-neg 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in b around inf 68.6%

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

      1. neg-mul-1 68.6%

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

      2. distribute-lft-neg-in 68.6%

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

      3. *-commutative 68.6%

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

   8. Simplified 68.6%

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

   9. Taylor expanded in b around 0 29.9%

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

       1. neg-mul-1 29.9%

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

       2. unsub-neg 29.9%

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

   11. Simplified 29.9%

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

   12. Taylor expanded in a around inf 27.7%

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

       1. mul-1-neg 27.7%

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

       2. associate-*r* 31.0%

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

       3. *-commutative 31.0%

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

       4. distribute-rgt-neg-in 31.0%

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

       5. distribute-rgt-neg-in 31.0%

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

   14. Simplified 31.0%

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

   if -6.5999999999999998e24 < a < 3.20000000000000006e185

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 62.7%

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

      1. mul-1-neg 62.7%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. *-commutative 62.7%

         \[\leadsto x \cdot e^{-\color{blue}{y \cdot t}} \]

      3. distribute-lft-neg-in 62.7%

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

   8. Simplified 62.7%

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

   9. Taylor expanded in y around 0 37.2%

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

       1. mul-1-neg 37.2%

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

       2. *-commutative 37.2%

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

       3. unsub-neg 37.2%

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

   11. Simplified 37.2%

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

   if 3.20000000000000006e185 < a

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0 92.0%

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

      1. +-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. associate-*r* 92.0%

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

      4. distribute-lft-out 92.0%

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

      5. mul-1-neg 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in b around inf 80.5%

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

      1. neg-mul-1 80.5%

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

      2. distribute-lft-neg-in 80.5%

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

      3. *-commutative 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in b around 0 26.3%

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

       1. neg-mul-1 26.3%

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

       2. unsub-neg 26.3%

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

   11. Simplified 26.3%

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

   12. Taylor expanded in a around inf 34.0%

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

       1. mul-1-neg 34.0%

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

       2. *-commutative 34.0%

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

       3. distribute-rgt-neg-in 34.0%

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

       4. distribute-rgt-neg-in 34.0%

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

   14. Simplified 34.0%

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

Alternative 11: 27.2% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-21} \lor \neg \left(y \leq 2.7 \cdot 10^{-103}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.8e-21) (not (<= y 2.7e-103))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.8e-21) || !(y <= 2.7e-103)) {
		tmp = a * (x * -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 <= (-2.8d-21)) .or. (.not. (y <= 2.7d-103))) then
        tmp = a * (x * -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 <= -2.8e-21) || !(y <= 2.7e-103)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.8e-21) or not (y <= 2.7e-103):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.8e-21) || !(y <= 2.7e-103))
		tmp = Float64(a * Float64(x * 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 <= -2.8e-21) || ~((y <= 2.7e-103)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.8e-21], N[Not[LessEqual[y, 2.7e-103]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000004e-21 or 2.7000000000000001e-103 < y

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-*r* 97.9%

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

      3. associate-*r* 97.9%

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

      4. distribute-lft-out 97.9%

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

      5. mul-1-neg 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in b around inf 43.0%

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

      1. neg-mul-1 43.0%

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

      2. distribute-lft-neg-in 43.0%

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

      3. *-commutative 43.0%

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

   8. Simplified 43.0%

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

   9. Taylor expanded in b around 0 14.8%

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

       1. neg-mul-1 14.8%

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

       2. unsub-neg 14.8%

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

   11. Simplified 14.8%

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

   12. Taylor expanded in a around inf 21.5%

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

       1. mul-1-neg 21.5%

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

       2. *-commutative 21.5%

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

       3. distribute-rgt-neg-in 21.5%

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

       4. distribute-rgt-neg-in 21.5%

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

   14. Simplified 21.5%

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

   if -2.80000000000000004e-21 < y < 2.7000000000000001e-103

   1. Initial program 93.6%

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

      1. fma-define 93.6%

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

      2. sub-neg 93.6%

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

      3. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 50.9%

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

   6. Taylor expanded in y around 0 39.1%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-21} \lor \neg \left(y \leq 2.7 \cdot 10^{-103}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.6% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-21}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3e-21) (* a (* x (- b))) (if (<= y 1e-103) x (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3e-21) {
		tmp = a * (x * -b);
	} else if (y <= 1e-103) {
		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 <= (-3d-21)) then
        tmp = a * (x * -b)
    else if (y <= 1d-103) 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 <= -3e-21) {
		tmp = a * (x * -b);
	} else if (y <= 1e-103) {
		tmp = x;
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3e-21:
		tmp = a * (x * -b)
	elif y <= 1e-103:
		tmp = x
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3e-21)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (y <= 1e-103)
		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 <= -3e-21)
		tmp = a * (x * -b);
	elseif (y <= 1e-103)
		tmp = x;
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3e-21], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-103], x, N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.99999999999999991e-21

   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. Taylor expanded in z around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. associate-*r* 98.5%

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

      4. distribute-lft-out 98.5%

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

      5. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf 34.1%

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

      1. neg-mul-1 34.1%

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

      2. distribute-lft-neg-in 34.1%

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

      3. *-commutative 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in b around 0 7.7%

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

       1. neg-mul-1 7.7%

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

       2. unsub-neg 7.7%

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

   11. Simplified 7.7%

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

   12. Taylor expanded in a around inf 13.4%

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

       1. mul-1-neg 13.4%

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

       2. *-commutative 13.4%

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

       3. distribute-rgt-neg-in 13.4%

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

       4. distribute-rgt-neg-in 13.4%

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

   14. Simplified 13.4%

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

   if -2.99999999999999991e-21 < y < 9.99999999999999958e-104

   1. Initial program 93.6%

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

      1. fma-define 93.6%

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

      2. sub-neg 93.6%

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

      3. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 50.9%

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

   6. Taylor expanded in y around 0 39.1%

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

   if 9.99999999999999958e-104 < y

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. associate-*r* 97.3%

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

      3. associate-*r* 97.3%

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

      4. distribute-lft-out 97.3%

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

      5. mul-1-neg 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in b around inf 51.0%

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

      1. neg-mul-1 51.0%

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

      2. distribute-lft-neg-in 51.0%

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

      3. *-commutative 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in b around 0 21.3%

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

       1. neg-mul-1 21.3%

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

       2. unsub-neg 21.3%

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

   11. Simplified 21.3%

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

   12. Taylor expanded in a around inf 28.8%

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

       1. mul-1-neg 28.8%

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

       2. associate-*r* 31.0%

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

       3. *-commutative 31.0%

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

       4. distribute-rgt-neg-in 31.0%

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

       5. distribute-rgt-neg-in 31.0%

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

   14. Simplified 31.0%

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

Alternative 13: 19.8% 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 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in a around 0 68.2%

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

6. Taylor expanded in y around 0 19.4%

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

Reproduce

herbie shell --seed 2024163 
(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))))))