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

Percentage Accurate: 96.5% → 99.6%

Time: 18.4s

Alternatives: 14

Speedup: 1.5×

• Could not converge on a ground truth

  1. reg0 = (bf "4.013635289541002876446627516198551884213e-226")
  2. reg1 = (bf "-2.958815503530752772267561966791156818549e299")
  3. reg2 = (bf "2.796128883160048311021985355622756496964e-168")
  4. reg3 = (bf "3.596926573525964747170413512873912142723e-292")
  5. reg4 = (bf "-2.710025223321858730513026335663020496271e-239")
  6. reg5 = (bf "9.851102624273762021172071434321328164144e155")

• 40.5% 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: 99.6% 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 95.6%

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

   1. fma-define 96.4%

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

   2. sub-neg 96.4%

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

   3. log1p-define 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0064:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -0.0064)
   (* x (pow (/ z (exp t)) y))
   (if (<= y 2.7e-5)
     (* x (exp (* a (- (- b) z))))
     (* x (exp (* y (- (log z) t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.0064) {
		tmp = x * pow((z / exp(t)), y);
	} else if (y <= 2.7e-5) {
		tmp = x * exp((a * (-b - z)));
	} else {
		tmp = x * exp((y * (log(z) - 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 (y <= (-0.0064d0)) then
        tmp = x * ((z / exp(t)) ** y)
    else if (y <= 2.7d-5) then
        tmp = x * exp((a * (-b - z)))
    else
        tmp = x * exp((y * (log(z) - 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 (y <= -0.0064) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else if (y <= 2.7e-5) {
		tmp = x * Math.exp((a * (-b - z)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -0.0064:
		tmp = x * math.pow((z / math.exp(t)), y)
	elif y <= 2.7e-5:
		tmp = x * math.exp((a * (-b - z)))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -0.0064)
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	elseif (y <= 2.7e-5)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -0.0064)
		tmp = x * ((z / exp(t)) ^ y);
	elseif (y <= 2.7e-5)
		tmp = x * exp((a * (-b - z)));
	else
		tmp = x * exp((y * (log(z) - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -0.0064], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-5], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00640000000000000031

   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. Step-by-step derivation

      1. fma-define 96.8%

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

      2. sub-neg 96.8%

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

      3. log1p-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in a around 0 89.1%

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

      1. pow1 89.1%

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

      2. pow-exp 89.1%

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

      3. pow-exp 89.1%

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

      4. *-commutative 89.1%

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

      5. exp-prod 89.1%

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

      6. exp-diff 89.1%

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

      7. add-exp-log 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. unpow1 89.2%

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

   9. Simplified 89.2%

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

   if -0.00640000000000000031 < y < 2.6999999999999999e-5

   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 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 y around 0 86.9%

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

      1. associate-*r* 86.9%

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

      2. mul-1-neg 86.9%

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

      3. +-commutative 86.9%

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

      4. distribute-lft-neg-in 86.9%

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

      5. distribute-rgt-neg-in 86.9%

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

   8. Simplified 86.9%

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

   if 2.6999999999999999e-5 < 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. Step-by-step derivation

      1. fma-define 98.3%

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

      2. sub-neg 98.3%

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

      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 86.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0064:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.0038) (not (<= y 2.75e-5)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.0038) || !(y <= 2.75e-5)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	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 <= (-0.0038d0)) .or. (.not. (y <= 2.75d-5))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp((a * (-b - z)))
    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 <= -0.0038) || !(y <= 2.75e-5)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.0038) or not (y <= 2.75e-5):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.0038) || !(y <= 2.75e-5))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(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 <= -0.0038) || ~((y <= 2.75e-5)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.0038], N[Not[LessEqual[y, 2.75e-5]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00379999999999999999 or 2.7500000000000001e-5 < y

   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 97.5%

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

      2. sub-neg 97.5%

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

      3. log1p-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in a around 0 88.0%

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

      1. pow1 88.0%

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

      2. pow-exp 88.0%

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

      3. pow-exp 88.0%

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

      4. *-commutative 88.0%

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

      5. exp-prod 88.0%

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

      6. exp-diff 88.0%

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

      7. add-exp-log 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. unpow1 88.1%

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

   9. Simplified 88.1%

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

   if -0.00379999999999999999 < y < 2.7500000000000001e-5

   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 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 y around 0 86.9%

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

      1. associate-*r* 86.9%

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

      2. mul-1-neg 86.9%

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

      3. +-commutative 86.9%

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

      4. distribute-lft-neg-in 86.9%

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

      5. distribute-rgt-neg-in 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% 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 95.6%

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

3. Taylor expanded in z around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-*r* 98.7%

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

   3. associate-*r* 98.7%

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

   4. distribute-lft-out 98.7%

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

   5. mul-1-neg 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00043:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.5e+15)
   (* x (pow z y))
   (if (<= y 0.00043)
     (* x (exp (* a (- (- b) z))))
     (if (<= y 2.6e+187)
       (* x (pow (/ z (+ t 1.0)) y))
       (* x (exp (* t (- y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e+15) {
		tmp = x * pow(z, y);
	} else if (y <= 0.00043) {
		tmp = x * exp((a * (-b - z)));
	} else if (y <= 2.6e+187) {
		tmp = x * pow((z / (t + 1.0)), y);
	} else {
		tmp = x * exp((t * -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 (y <= (-6.5d+15)) then
        tmp = x * (z ** y)
    else if (y <= 0.00043d0) then
        tmp = x * exp((a * (-b - z)))
    else if (y <= 2.6d+187) then
        tmp = x * ((z / (t + 1.0d0)) ** y)
    else
        tmp = x * exp((t * -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 (y <= -6.5e+15) {
		tmp = x * Math.pow(z, y);
	} else if (y <= 0.00043) {
		tmp = x * Math.exp((a * (-b - z)));
	} else if (y <= 2.6e+187) {
		tmp = x * Math.pow((z / (t + 1.0)), y);
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.5e+15:
		tmp = x * math.pow(z, y)
	elif y <= 0.00043:
		tmp = x * math.exp((a * (-b - z)))
	elif y <= 2.6e+187:
		tmp = x * math.pow((z / (t + 1.0)), y)
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.5e+15)
		tmp = Float64(x * (z ^ y));
	elseif (y <= 0.00043)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	elseif (y <= 2.6e+187)
		tmp = Float64(x * (Float64(z / Float64(t + 1.0)) ^ y));
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.5e+15)
		tmp = x * (z ^ y);
	elseif (y <= 0.00043)
		tmp = x * exp((a * (-b - z)));
	elseif (y <= 2.6e+187)
		tmp = x * ((z / (t + 1.0)) ^ y);
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.5e+15], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00043], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+187], N[(x * N[Power[N[(z / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.5e15

   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 96.7%

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

      2. sub-neg 96.7%

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

      3. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in a around 0 88.5%

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

   6. Taylor expanded in t around 0 65.5%

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

   if -6.5e15 < y < 4.29999999999999989e-4

   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. Add Preprocessing

   3. Taylor expanded in z around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. distribute-lft-out 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. mul-1-neg 85.9%

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

      3. +-commutative 85.9%

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

      4. distribute-lft-neg-in 85.9%

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

      5. distribute-rgt-neg-in 85.9%

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

   8. Simplified 85.9%

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

   if 4.29999999999999989e-4 < y < 2.5999999999999999e187

   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. Step-by-step derivation

      1. fma-define 97.6%

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

      2. sub-neg 97.6%

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

      3. log1p-define 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 85.9%

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

      1. pow1 85.9%

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

      2. pow-exp 85.8%

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

      3. pow-exp 85.9%

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

      4. *-commutative 85.9%

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

      5. exp-prod 85.9%

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

      6. exp-diff 85.9%

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

      7. add-exp-log 85.9%

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

   7. Applied egg-rr 85.9%

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

      1. unpow1 85.9%

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

   9. Simplified 85.9%

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

   10. Taylor expanded in t around 0 81.2%

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

   if 2.5999999999999999e187 < 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 a around 0 89.1%

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

   6. Taylor expanded in t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   8. Simplified 94.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00043:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00036:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -1.65e+16)
     t_1
     (if (<= y 0.00036)
       (* x (exp (* a (- (- b) z))))
       (if (<= y 8.8e+186) t_1 (* x (exp (* t (- y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -1.65e+16) {
		tmp = t_1;
	} else if (y <= 0.00036) {
		tmp = x * exp((a * (-b - z)));
	} else if (y <= 8.8e+186) {
		tmp = t_1;
	} else {
		tmp = x * exp((t * -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) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-1.65d+16)) then
        tmp = t_1
    else if (y <= 0.00036d0) then
        tmp = x * exp((a * (-b - z)))
    else if (y <= 8.8d+186) then
        tmp = t_1
    else
        tmp = x * exp((t * -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 t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -1.65e+16) {
		tmp = t_1;
	} else if (y <= 0.00036) {
		tmp = x * Math.exp((a * (-b - z)));
	} else if (y <= 8.8e+186) {
		tmp = t_1;
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -1.65e+16:
		tmp = t_1
	elif y <= 0.00036:
		tmp = x * math.exp((a * (-b - z)))
	elif y <= 8.8e+186:
		tmp = t_1
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -1.65e+16)
		tmp = t_1;
	elseif (y <= 0.00036)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	elseif (y <= 8.8e+186)
		tmp = t_1;
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -1.65e+16)
		tmp = t_1;
	elseif (y <= 0.00036)
		tmp = x * exp((a * (-b - z)));
	elseif (y <= 8.8e+186)
		tmp = t_1;
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65e+16], t$95$1, If[LessEqual[y, 0.00036], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.8e+186], t$95$1, N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.65e16 or 3.60000000000000023e-4 < y < 8.7999999999999993e186

   1. Initial program 95.1%

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

      1. fma-define 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-define 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in a around 0 87.4%

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

   6. Taylor expanded in t around 0 71.8%

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

   if -1.65e16 < y < 3.60000000000000023e-4

   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. Add Preprocessing

   3. Taylor expanded in z around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. distribute-lft-out 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. mul-1-neg 85.9%

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

      3. +-commutative 85.9%

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

      4. distribute-lft-neg-in 85.9%

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

      5. distribute-rgt-neg-in 85.9%

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

   8. Simplified 85.9%

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

   if 8.7999999999999993e186 < 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 a around 0 89.1%

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

   6. Taylor expanded in t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00036:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+186}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -5.2e+15)
     t_1
     (if (<= y 0.00045)
       (* x (exp (* a (- b))))
       (if (<= y 2.9e+185) t_1 (* x (exp (* t (- y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -5.2e+15) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * exp((a * -b));
	} else if (y <= 2.9e+185) {
		tmp = t_1;
	} else {
		tmp = x * exp((t * -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) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-5.2d+15)) then
        tmp = t_1
    else if (y <= 0.00045d0) then
        tmp = x * exp((a * -b))
    else if (y <= 2.9d+185) then
        tmp = t_1
    else
        tmp = x * exp((t * -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 t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -5.2e+15) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * Math.exp((a * -b));
	} else if (y <= 2.9e+185) {
		tmp = t_1;
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -5.2e+15:
		tmp = t_1
	elif y <= 0.00045:
		tmp = x * math.exp((a * -b))
	elif y <= 2.9e+185:
		tmp = t_1
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -5.2e+15)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (y <= 2.9e+185)
		tmp = t_1;
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -5.2e+15)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = x * exp((a * -b));
	elseif (y <= 2.9e+185)
		tmp = t_1;
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+15], t$95$1, If[LessEqual[y, 0.00045], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+185], t$95$1, N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e15 or 4.4999999999999999e-4 < y < 2.89999999999999988e185

   1. Initial program 95.1%

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

      1. fma-define 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-define 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in a around 0 87.4%

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

   6. Taylor expanded in t around 0 71.8%

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

   if -5.2e15 < y < 4.4999999999999999e-4

   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. Add Preprocessing

   3. Taylor expanded in z around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. distribute-lft-out 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. mul-1-neg 85.9%

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

      3. +-commutative 85.9%

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

      4. distribute-lft-neg-in 85.9%

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

      5. distribute-rgt-neg-in 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in z around 0 79.3%

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

       1. mul-1-neg 79.3%

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

       2. *-commutative 79.3%

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

       3. distribute-rgt-neg-in 79.3%

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

   11. Simplified 79.3%

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

   if 2.89999999999999988e185 < 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 a around 0 89.1%

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

   6. Taylor expanded in t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+185}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-14} \lor \neg \left(b \leq 1.8 \cdot 10^{-108}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -9.5e-14) (not (<= b 1.8e-108)))
   (* 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 ((b <= -9.5e-14) || !(b <= 1.8e-108)) {
		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 ((b <= (-9.5d-14)) .or. (.not. (b <= 1.8d-108))) 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 ((b <= -9.5e-14) || !(b <= 1.8e-108)) {
		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 (b <= -9.5e-14) or not (b <= 1.8e-108):
		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 ((b <= -9.5e-14) || !(b <= 1.8e-108))
		tmp = Float64(x * exp(Float64(a * Float64(-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 ((b <= -9.5e-14) || ~((b <= 1.8e-108)))
		tmp = x * exp((a * -b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -9.5e-14], N[Not[LessEqual[b, 1.8e-108]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.4999999999999999e-14 or 1.8e-108 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0 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 y around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

      4. distribute-lft-neg-in 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in z around 0 78.8%

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

       1. mul-1-neg 78.8%

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

       2. *-commutative 78.8%

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

       3. distribute-rgt-neg-in 78.8%

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

   11. Simplified 78.8%

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

   if -9.4999999999999999e-14 < b < 1.8e-108

   1. Initial program 92.6%

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

      1. fma-define 92.6%

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

      2. sub-neg 92.6%

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

      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 90.1%

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

   6. Taylor expanded in t around 0 69.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-14} \lor \neg \left(b \leq 1.8 \cdot 10^{-108}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18} \lor \neg \left(y \leq 1.46 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.7e-18) (not (<= y 1.46e-36)))
   (* x (pow z y))
   (* x (- 1.0 (* a (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.7e-18) || !(y <= 1.46e-36)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (z + 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 <= (-1.7d-18)) .or. (.not. (y <= 1.46d-36))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 - (a * (z + 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 <= -1.7e-18) || !(y <= 1.46e-36)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.7e-18) or not (y <= 1.46e-36):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 - (a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.7e-18) || !(y <= 1.46e-36))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.7e-18) || ~((y <= 1.46e-36)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 - (a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.7e-18], N[Not[LessEqual[y, 1.46e-36]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000001e-18 or 1.4599999999999999e-36 < y

   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. Step-by-step derivation

      1. fma-define 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in a around 0 84.3%

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

   6. Taylor expanded in t around 0 64.4%

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

   if -1.70000000000000001e-18 < y < 1.4599999999999999e-36

   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 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 y around 0 87.7%

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

      1. associate-*r* 87.7%

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

      2. mul-1-neg 87.7%

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

      3. +-commutative 87.7%

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

      4. distribute-lft-neg-in 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in a around 0 50.2%

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

       1. associate-*r* 50.2%

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

       2. mul-1-neg 50.2%

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

       3. +-commutative 50.2%

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

       4. distribute-lft-neg-in 50.2%

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

       5. unsub-neg 50.2%

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

       6. +-commutative 50.2%

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

   11. Simplified 50.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18} \lor \neg \left(y \leq 1.46 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.0% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.7e-18)
   (* t (* x (- y)))
   (if (<= y 6.2e-24) (* x (- 1.0 (* a (+ z b)))) (* x (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.7e-18) {
		tmp = t * (x * -y);
	} else if (y <= 6.2e-24) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = x * (t * -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 (y <= (-1.7d-18)) then
        tmp = t * (x * -y)
    else if (y <= 6.2d-24) then
        tmp = x * (1.0d0 - (a * (z + b)))
    else
        tmp = x * (t * -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 (y <= -1.7e-18) {
		tmp = t * (x * -y);
	} else if (y <= 6.2e-24) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.7e-18:
		tmp = t * (x * -y)
	elif y <= 6.2e-24:
		tmp = x * (1.0 - (a * (z + b)))
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.7e-18)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 6.2e-24)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.7e-18)
		tmp = t * (x * -y);
	elseif (y <= 6.2e-24)
		tmp = x * (1.0 - (a * (z + b)));
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.7e-18], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-24], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000001e-18

   1. Initial program 94.1%

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

      1. fma-define 95.6%

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

      2. sub-neg 95.6%

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

      3. log1p-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in a around 0 85.7%

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

   6. Taylor expanded in t around inf 60.6%

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

      1. mul-1-neg 60.6%

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

      2. distribute-lft-neg-out 60.6%

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

      3. *-commutative 60.6%

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

   8. Simplified 60.6%

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

   9. Taylor expanded in y around 0 13.7%

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

       1. mul-1-neg 13.7%

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

       2. unsub-neg 13.7%

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

       3. *-commutative 13.7%

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

       4. associate-*l* 12.6%

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

   11. Simplified 12.6%

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

   12. Taylor expanded in y around inf 15.1%

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

       1. mul-1-neg 15.1%

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

       2. distribute-rgt-neg-in 15.1%

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

       3. distribute-rgt-neg-in 15.1%

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

   14. Simplified 15.1%

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

   if -1.70000000000000001e-18 < y < 6.2000000000000001e-24

   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. 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 y around 0 87.3%

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

      1. associate-*r* 87.3%

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

      2. mul-1-neg 87.3%

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

      3. +-commutative 87.3%

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

      4. distribute-lft-neg-in 87.3%

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

      5. distribute-rgt-neg-in 87.3%

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

   8. Simplified 87.3%

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

   9. Taylor expanded in a around 0 51.0%

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

       1. associate-*r* 51.0%

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

       2. mul-1-neg 51.0%

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

       3. +-commutative 51.0%

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

       4. distribute-lft-neg-in 51.0%

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

       5. unsub-neg 51.0%

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

       6. +-commutative 51.0%

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

   11. Simplified 51.0%

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

   if 6.2000000000000001e-24 < y

   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 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 81.9%

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

   6. Taylor expanded in t around inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. distribute-lft-neg-out 58.3%

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

      3. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in y around 0 15.4%

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

       1. mul-1-neg 15.4%

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

       2. unsub-neg 15.4%

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

       3. *-commutative 15.4%

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

       4. associate-*l* 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in y around inf 21.0%

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

       1. mul-1-neg 21.0%

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

       2. *-commutative 21.0%

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

       3. associate-*r* 21.1%

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

       4. distribute-rgt-neg-out 21.1%

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

       5. distribute-lft-neg-in 21.1%

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

   14. Simplified 21.1%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-12} \lor \neg \left(y \leq 2.6 \cdot 10^{-25}\right):\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.8e-12) (not (<= y 2.6e-25))) (* t (* x (- y))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.8e-12) || !(y <= 2.6e-25)) {
		tmp = t * (x * -y);
	} 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 <= (-6.8d-12)) .or. (.not. (y <= 2.6d-25))) then
        tmp = t * (x * -y)
    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 <= -6.8e-12) || !(y <= 2.6e-25)) {
		tmp = t * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.8e-12) or not (y <= 2.6e-25):
		tmp = t * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.8e-12) || !(y <= 2.6e-25))
		tmp = Float64(t * Float64(x * Float64(-y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.8e-12) || ~((y <= 2.6e-25)))
		tmp = t * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.8e-12], N[Not[LessEqual[y, 2.6e-25]], $MachinePrecision]], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.8000000000000001e-12 or 2.6e-25 < y

   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. 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 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in a around 0 85.1%

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

   6. Taylor expanded in t around inf 60.4%

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

      1. mul-1-neg 60.4%

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

      2. distribute-lft-neg-out 60.4%

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

      3. *-commutative 60.4%

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

   8. Simplified 60.4%

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

   9. Taylor expanded in y around 0 14.7%

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

       1. mul-1-neg 14.7%

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

       2. unsub-neg 14.7%

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

       3. *-commutative 14.7%

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

       4. associate-*l* 14.2%

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

   11. Simplified 14.2%

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

   12. Taylor expanded in y around inf 18.2%

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

       1. mul-1-neg 18.2%

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

       2. distribute-rgt-neg-in 18.2%

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

       3. distribute-rgt-neg-in 18.2%

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

   14. Simplified 18.2%

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

   if -6.8000000000000001e-12 < y < 2.6e-25

   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 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 54.9%

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

   6. Taylor expanded in y around 0 35.6%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-12} \lor \neg \left(y \leq 2.6 \cdot 10^{-25}\right):\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.38e-11) (* t (* x (- y))) (if (<= y 6e-24) x (* x (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.38e-11) {
		tmp = t * (x * -y);
	} else if (y <= 6e-24) {
		tmp = x;
	} else {
		tmp = x * (t * -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 (y <= (-1.38d-11)) then
        tmp = t * (x * -y)
    else if (y <= 6d-24) then
        tmp = x
    else
        tmp = x * (t * -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 (y <= -1.38e-11) {
		tmp = t * (x * -y);
	} else if (y <= 6e-24) {
		tmp = x;
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.38e-11:
		tmp = t * (x * -y)
	elif y <= 6e-24:
		tmp = x
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.38e-11)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 6e-24)
		tmp = x;
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.38e-11)
		tmp = t * (x * -y);
	elseif (y <= 6e-24)
		tmp = x;
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.38e-11], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-24], x, N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.38e-11

   1. Initial program 94.0%

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

      1. fma-define 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 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in a around 0 88.2%

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

   6. Taylor expanded in t around inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-lft-neg-out 62.4%

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

      3. *-commutative 62.4%

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

   8. Simplified 62.4%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. mul-1-neg 14.1%

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

       2. unsub-neg 14.1%

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

       3. *-commutative 14.1%

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

       4. associate-*l* 13.0%

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

   11. Simplified 13.0%

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

   12. Taylor expanded in y around inf 15.5%

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

       1. mul-1-neg 15.5%

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

       2. distribute-rgt-neg-in 15.5%

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

       3. distribute-rgt-neg-in 15.5%

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

   14. Simplified 15.5%

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

   if -1.38e-11 < y < 5.99999999999999991e-24

   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 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 54.9%

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

   6. Taylor expanded in y around 0 35.6%

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

   if 5.99999999999999991e-24 < y

   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 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 81.9%

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

   6. Taylor expanded in t around inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. distribute-lft-neg-out 58.3%

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

      3. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in y around 0 15.4%

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

       1. mul-1-neg 15.4%

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

       2. unsub-neg 15.4%

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

       3. *-commutative 15.4%

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

       4. associate-*l* 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in y around inf 21.0%

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

       1. mul-1-neg 21.0%

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

       2. *-commutative 21.0%

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

       3. associate-*r* 21.1%

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

       4. distribute-rgt-neg-out 21.1%

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

       5. distribute-lft-neg-in 21.1%

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

   14. Simplified 21.1%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 6.2e-24) (- x (* a (* x b))) (* x (* t (- y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 6.2e-24) {
		tmp = x - (a * (x * b));
	} else {
		tmp = x * (t * -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 (y <= 6.2d-24) then
        tmp = x - (a * (x * b))
    else
        tmp = x * (t * -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 (y <= 6.2e-24) {
		tmp = x - (a * (x * b));
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 6.2e-24:
		tmp = x - (a * (x * b))
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 6.2e-24)
		tmp = Float64(x - Float64(a * Float64(x * b)));
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 6.2e-24)
		tmp = x - (a * (x * b));
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 6.2e-24], N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000001e-24

   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. 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 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 64.5%

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

   6. Taylor expanded in y around 0 34.0%

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

      1. sub-neg 34.0%

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

      2. log1p-undefine 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in z around 0 34.0%

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

       1. mul-1-neg 34.0%

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

       2. unsub-neg 34.0%

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

   11. Simplified 34.0%

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

   if 6.2000000000000001e-24 < y

   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 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 81.9%

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

   6. Taylor expanded in t around inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. distribute-lft-neg-out 58.3%

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

      3. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in y around 0 15.4%

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

       1. mul-1-neg 15.4%

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

       2. unsub-neg 15.4%

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

       3. *-commutative 15.4%

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

       4. associate-*l* 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in y around inf 21.0%

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

       1. mul-1-neg 21.0%

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

       2. *-commutative 21.0%

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

       3. associate-*r* 21.1%

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

       4. distribute-rgt-neg-out 21.1%

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

       5. distribute-lft-neg-in 21.1%

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

   14. Simplified 21.1%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 19.5% 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 95.6%

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

   1. fma-define 96.4%

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

   2. sub-neg 96.4%

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

   3. log1p-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in a around 0 70.4%

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

6. Taylor expanded in y around 0 19.2%

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

Reproduce

herbie shell --seed 2024137 
(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))))))