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

Percentage Accurate: 96.4% → 99.6%

Time: 13.1s

Alternatives: 15

Speedup: 1.0×

• 20.2% 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: 96.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. +-commutative 95.4%

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

   2. fma-def 95.8%

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

   3. sub-neg 95.8%

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

   4. log1p-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 96.4% 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]

Derivation

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. Final simplification 95.4%

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

Alternative 3: 87.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -28000000000000 \lor \neg \left(y \leq 10^{+20}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -28000000000000.0) (not (<= y 1e+20)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (- z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -28000000000000.0) || !(y <= 1e+20)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((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 <= (-28000000000000.0d0)) .or. (.not. (y <= 1d+20))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((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 <= -28000000000000.0) || !(y <= 1e+20)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (-z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -28000000000000.0) or not (y <= 1e+20):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (-z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -28000000000000.0) || !(y <= 1e+20))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -28000000000000.0) || ~((y <= 1e+20)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (-z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -28000000000000.0], N[Not[LessEqual[y, 1e+20]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e13 or 1e20 < y

   1. Initial program 97.6%

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

   2. Taylor expanded in y around inf 88.8%

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

   if -2.8e13 < y < 1e20

   1. Initial program 93.4%

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

   2. Taylor expanded in y around 0 79.8%

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

      1. sub-neg 79.8%

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

      2. mul-1-neg 79.8%

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

      3. log1p-def 88.5%

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

      4. mul-1-neg 88.5%

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

   4. Simplified 88.5%

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

   5. Taylor expanded in z around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. associate-*r* 88.5%

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

      3. associate-*r* 88.5%

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

      4. distribute-lft-out 88.5%

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

      5. mul-1-neg 88.5%

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

   7. Simplified 88.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28000000000000 \lor \neg \left(y \leq 10^{+20}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]

Alternative 4: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14000000000000 \lor \neg \left(y \leq 7 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -14000000000000.0) (not (<= y 7e+19)))
   (* x (pow z y))
   (* x (exp (* a (- (- z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -14000000000000.0) || !(y <= 7e+19)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((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 <= (-14000000000000.0d0)) .or. (.not. (y <= 7d+19))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((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 <= -14000000000000.0) || !(y <= 7e+19)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * (-z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -14000000000000.0) or not (y <= 7e+19):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * (-z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -14000000000000.0) || !(y <= 7e+19))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -14000000000000.0) || ~((y <= 7e+19)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * (-z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -14000000000000.0], N[Not[LessEqual[y, 7e+19]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e13 or 7e19 < y

   1. Initial program 97.6%

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

   2. Taylor expanded in y around inf 88.8%

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

   3. Taylor expanded in t around 0 74.4%

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

   if -1.4e13 < y < 7e19

   1. Initial program 93.4%

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

   2. Taylor expanded in y around 0 79.8%

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

      1. sub-neg 79.8%

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

      2. mul-1-neg 79.8%

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

      3. log1p-def 88.5%

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

      4. mul-1-neg 88.5%

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

   4. Simplified 88.5%

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

   5. Taylor expanded in z around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. associate-*r* 88.5%

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

      3. associate-*r* 88.5%

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

      4. distribute-lft-out 88.5%

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

      5. mul-1-neg 88.5%

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

   7. Simplified 88.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14000000000000 \lor \neg \left(y \leq 7 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]

Alternative 5: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -19000000000000 \lor \neg \left(y \leq 0.038\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -19000000000000.0) (not (<= y 0.038)))
   (* x (pow z y))
   (* x (exp (* b (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -19000000000000.0) || !(y <= 0.038)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((b * -a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-19000000000000.0d0)) .or. (.not. (y <= 0.038d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((b * -a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -19000000000000.0) || !(y <= 0.038)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((b * -a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -19000000000000.0) or not (y <= 0.038):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((b * -a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -19000000000000.0) || !(y <= 0.038))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(b * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -19000000000000.0) || ~((y <= 0.038)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((b * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -19000000000000.0], N[Not[LessEqual[y, 0.038]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9e13 or 0.0379999999999999991 < 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. Taylor expanded in y around inf 87.5%

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

   3. Taylor expanded in t around 0 74.2%

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

   if -1.9e13 < y < 0.0379999999999999991

   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. Taylor expanded in b around inf 80.7%

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

      1. associate-*r* 80.7%

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

      2. neg-mul-1 80.7%

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

      3. *-commutative 80.7%

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

   4. Simplified 80.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19000000000000 \lor \neg \left(y \leq 0.038\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \end{array} \]

Alternative 6: 54.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.1e+169) (* x (- 1.0 (* y t))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.1e+169) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-2.1d+169)) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.1e+169) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.1e+169:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.1e+169)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.1e+169)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.1e+169], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e169

   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. Taylor expanded in t around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. distribute-rgt-neg-out 89.8%

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

   4. Simplified 89.8%

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

   5. Taylor expanded in y around 0 25.9%

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

      1. +-commutative 25.9%

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

      2. mul-1-neg 25.9%

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

      3. unsub-neg 25.9%

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

      4. associate-*r* 35.1%

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

      5. *-commutative 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in x around 0 35.1%

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

   if -2.1000000000000001e169 < t

   1. Initial program 94.8%

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

   2. Taylor expanded in y around inf 67.5%

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

   3. Taylor expanded in t around 0 59.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 7: 31.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.00086:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.05e-73)
   (* t (* x (- y)))
   (if (<= y 0.00086) (* x (- 1.0 (* a b))) (* y (* x (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.05e-73) {
		tmp = t * (x * -y);
	} else if (y <= 0.00086) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (x * -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 <= (-1.05d-73)) then
        tmp = t * (x * -y)
    else if (y <= 0.00086d0) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = y * (x * -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 <= -1.05e-73) {
		tmp = t * (x * -y);
	} else if (y <= 0.00086) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.05e-73:
		tmp = t * (x * -y)
	elif y <= 0.00086:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.05e-73)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 0.00086)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.05e-73)
		tmp = t * (x * -y);
	elseif (y <= 0.00086)
		tmp = x * (1.0 - (a * b));
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.05e-73], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00086], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0499999999999999e-73

   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. Taylor expanded in t around inf 56.7%

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

      1. mul-1-neg 56.7%

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

      2. distribute-rgt-neg-out 56.7%

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

   4. Simplified 56.7%

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

   5. Taylor expanded in y around 0 16.3%

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

      1. +-commutative 16.3%

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

      2. mul-1-neg 16.3%

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

      3. unsub-neg 16.3%

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

      4. associate-*r* 17.5%

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

      5. *-commutative 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 17.3%

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

      1. expm1-log1p-u 10.4%

         \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(t \cdot x\right)\right)\right)} \]

      2. expm1-udef 17.4%

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

      3. *-commutative 17.4%

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

      4. associate-*l* 19.7%

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

   10. Applied egg-rr 19.7%

       \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 12.8%

          \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)\right)} \]

       2. expm1-log1p 19.6%

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

   12. Simplified 19.6%

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

   if -1.0499999999999999e-73 < y < 8.59999999999999979e-4

   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. Taylor expanded in b around inf 85.6%

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

      1. associate-*r* 85.6%

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

      2. neg-mul-1 85.6%

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

      3. *-commutative 85.6%

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

   4. Simplified 85.6%

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

   5. Taylor expanded in b around 0 46.0%

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

      1. +-commutative 46.0%

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

      2. mul-1-neg 46.0%

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

      3. unsub-neg 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in x around 0 46.8%

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

   if 8.59999999999999979e-4 < y

   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. Taylor expanded in t around inf 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-rgt-neg-out 54.8%

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

   4. Simplified 54.8%

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

   5. Taylor expanded in y around 0 11.7%

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

      1. +-commutative 11.7%

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

      2. mul-1-neg 11.7%

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

      3. unsub-neg 11.7%

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

      4. associate-*r* 20.5%

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

      5. *-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in y around inf 28.2%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.00086:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 8: 32.4% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.8e+141)
   (* x (- 1.0 (* a b)))
   (if (<= a 2.2e+133) (* x (- 1.0 (* y t))) (* a (- (* x b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.8e+141) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 2.2e+133) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.8d+141)) then
        tmp = x * (1.0d0 - (a * b))
    else if (a <= 2.2d+133) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = a * -(x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.8e+141) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 2.2e+133) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -5.8e+141:
		tmp = x * (1.0 - (a * b))
	elif a <= 2.2e+133:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = a * -(x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.8e+141)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (a <= 2.2e+133)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(a * Float64(-Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -5.8e+141)
		tmp = x * (1.0 - (a * b));
	elseif (a <= 2.2e+133)
		tmp = x * (1.0 - (y * t));
	else
		tmp = a * -(x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.8e+141], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.2e+133], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.80000000000000013e141

   1. Initial program 90.1%

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

   2. Taylor expanded in b around inf 78.6%

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

      1. associate-*r* 78.6%

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

      2. neg-mul-1 78.6%

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

      3. *-commutative 78.6%

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

   4. Simplified 78.6%

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

   5. Taylor expanded in b around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. mul-1-neg 30.0%

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

      3. unsub-neg 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in x around 0 36.3%

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

   if -5.80000000000000013e141 < a < 2.2e133

   1. Initial program 99.5%

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

   2. Taylor expanded in t around inf 64.9%

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

      1. mul-1-neg 64.9%

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

      2. distribute-rgt-neg-out 64.9%

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

   4. Simplified 64.9%

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

   5. Taylor expanded in y around 0 31.1%

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

      1. +-commutative 31.1%

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

      2. mul-1-neg 31.1%

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

      3. unsub-neg 31.1%

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

      4. associate-*r* 34.3%

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

      5. *-commutative 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in x around 0 34.3%

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

   if 2.2e133 < a

   1. Initial program 84.0%

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

   2. Taylor expanded in b around inf 70.2%

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

      1. associate-*r* 70.2%

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

      2. neg-mul-1 70.2%

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

      3. *-commutative 70.2%

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

   4. Simplified 70.2%

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

   5. Taylor expanded in b around 0 20.8%

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

      1. +-commutative 20.8%

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

      2. mul-1-neg 20.8%

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

      3. unsub-neg 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in a around inf 30.5%

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

      1. neg-mul-1 30.5%

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

      2. distribute-rgt-neg-in 30.5%

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

      3. distribute-lft-neg-in 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \]

Alternative 9: 33.1% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.9 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.9e+19)
   (* x (- 1.0 (* a (+ z b))))
   (if (<= a 5.2e+134) (* x (- 1.0 (* y t))) (* a (- (* x b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.9e+19) {
		tmp = x * (1.0 - (a * (z + b)));
	} else if (a <= 5.2e+134) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.9d+19)) then
        tmp = x * (1.0d0 - (a * (z + b)))
    else if (a <= 5.2d+134) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = a * -(x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.9e+19) {
		tmp = x * (1.0 - (a * (z + b)));
	} else if (a <= 5.2e+134) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6.9e+19:
		tmp = x * (1.0 - (a * (z + b)))
	elif a <= 5.2e+134:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = a * -(x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6.9e+19)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	elseif (a <= 5.2e+134)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(a * Float64(-Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6.9e+19)
		tmp = x * (1.0 - (a * (z + b)));
	elseif (a <= 5.2e+134)
		tmp = x * (1.0 - (y * t));
	else
		tmp = a * -(x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6.9e+19], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.2e+134], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9e19

   1. Initial program 94.8%

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

   2. Taylor expanded in y around 0 66.7%

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

      1. sub-neg 66.7%

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

      2. mul-1-neg 66.7%

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

      3. log1p-def 80.2%

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

      4. mul-1-neg 80.2%

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

   4. Simplified 80.2%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. associate-*r* 80.2%

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

      3. associate-*r* 80.2%

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

      4. distribute-lft-out 80.2%

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

      5. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in a around 0 37.6%

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

      1. associate-*r* 37.6%

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

      2. +-commutative 37.6%

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

      3. associate-*r* 37.6%

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

      4. neg-mul-1 37.6%

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

      5. unsub-neg 37.6%

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

      6. +-commutative 37.6%

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

   10. Simplified 37.6%

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

   if -6.9e19 < a < 5.2000000000000003e134

   1. Initial program 99.4%

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

   2. Taylor expanded in t around inf 66.4%

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

      1. mul-1-neg 66.4%

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

      2. distribute-rgt-neg-out 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in y around 0 31.0%

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

      1. +-commutative 31.0%

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

      2. mul-1-neg 31.0%

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

      3. unsub-neg 31.0%

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

      4. associate-*r* 34.7%

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

      5. *-commutative 34.7%

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

   7. Simplified 34.7%

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

   8. Taylor expanded in x around 0 34.7%

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

   if 5.2000000000000003e134 < a

   1. Initial program 84.0%

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

   2. Taylor expanded in b around inf 70.2%

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

      1. associate-*r* 70.2%

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

      2. neg-mul-1 70.2%

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

      3. *-commutative 70.2%

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

   4. Simplified 70.2%

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

   5. Taylor expanded in b around 0 20.8%

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

      1. +-commutative 20.8%

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

      2. mul-1-neg 20.8%

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

      3. unsub-neg 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in a around inf 30.5%

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

      1. neg-mul-1 30.5%

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

      2. distribute-rgt-neg-in 30.5%

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

      3. distribute-lft-neg-in 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.9 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \]

Alternative 10: 27.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-97} \lor \neg \left(y \leq 3.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5e-97) (not (<= y 3.4e-8))) (* a (- (* x b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-97) || !(y <= 3.4e-8)) {
		tmp = a * -(x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5d-97)) .or. (.not. (y <= 3.4d-8))) then
        tmp = a * -(x * b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-97) || !(y <= 3.4e-8)) {
		tmp = a * -(x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5e-97) or not (y <= 3.4e-8):
		tmp = a * -(x * b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5e-97) || !(y <= 3.4e-8))
		tmp = Float64(a * Float64(-Float64(x * b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5e-97) || ~((y <= 3.4e-8)))
		tmp = a * -(x * b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5e-97], N[Not[LessEqual[y, 3.4e-8]], $MachinePrecision]], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999995e-97 or 3.4e-8 < y

   1. Initial program 94.8%

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

   2. Taylor expanded in b around inf 37.1%

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

      1. associate-*r* 37.1%

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

      2. neg-mul-1 37.1%

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

      3. *-commutative 37.1%

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

   4. Simplified 37.1%

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

   5. Taylor expanded in b around 0 10.3%

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

      1. +-commutative 10.3%

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

      2. mul-1-neg 10.3%

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

      3. unsub-neg 10.3%

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

   7. Simplified 10.3%

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

   8. Taylor expanded in a around inf 17.2%

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

      1. neg-mul-1 17.2%

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

      2. distribute-rgt-neg-in 17.2%

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

      3. distribute-lft-neg-in 17.2%

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

   10. Simplified 17.2%

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

   if -4.9999999999999995e-97 < y < 3.4e-8

   1. Initial program 96.3%

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

   2. Taylor expanded in b around inf 86.2%

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

      1. associate-*r* 86.2%

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

      2. neg-mul-1 86.2%

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

      3. *-commutative 86.2%

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

   4. Simplified 86.2%

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

   5. Taylor expanded in b around 0 38.0%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-97} \lor \neg \left(y \leq 3.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 27.0% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-112} \lor \neg \left(y \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(-y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.5e-112) (not (<= y 2.7e-13))) (* x (- (* y t))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.5e-112) || !(y <= 2.7e-13)) {
		tmp = x * -(y * t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-8.5d-112)) .or. (.not. (y <= 2.7d-13))) then
        tmp = x * -(y * t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.5e-112) || !(y <= 2.7e-13)) {
		tmp = x * -(y * t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.5e-112) or not (y <= 2.7e-13):
		tmp = x * -(y * t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.5e-112) || !(y <= 2.7e-13))
		tmp = Float64(x * Float64(-Float64(y * t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -8.5e-112) || ~((y <= 2.7e-13)))
		tmp = x * -(y * t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.5e-112], N[Not[LessEqual[y, 2.7e-13]], $MachinePrecision]], N[(x * (-N[(y * t), $MachinePrecision])), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -8.49999999999999992e-112 or 2.70000000000000011e-13 < y

   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. Taylor expanded in t around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. distribute-rgt-neg-out 53.5%

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

   4. Simplified 53.5%

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

   5. Taylor expanded in y around 0 14.6%

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

      1. +-commutative 14.6%

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

      2. mul-1-neg 14.6%

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

      3. unsub-neg 14.6%

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

      4. associate-*r* 18.6%

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

      5. *-commutative 18.6%

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

   7. Simplified 18.6%

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

   8. Taylor expanded in y around inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. distribute-rgt-neg-in 21.3%

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

      5. distribute-lft-neg-in 21.3%

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

   10. Simplified 21.3%

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

   if -8.49999999999999992e-112 < y < 2.70000000000000011e-13

   1. Initial program 96.1%

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

   2. Taylor expanded in b around inf 85.4%

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

      1. associate-*r* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in b around 0 39.0%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-112} \lor \neg \left(y \leq 2.7 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(-y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 26.8% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-112}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.5e-112)
   (* t (* x (- y)))
   (if (<= y 1.06e-12) x (* x (- (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e-112) {
		tmp = t * (x * -y);
	} else if (y <= 1.06e-12) {
		tmp = x;
	} else {
		tmp = x * -(y * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-8.5d-112)) then
        tmp = t * (x * -y)
    else if (y <= 1.06d-12) then
        tmp = x
    else
        tmp = x * -(y * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e-112) {
		tmp = t * (x * -y);
	} else if (y <= 1.06e-12) {
		tmp = x;
	} else {
		tmp = x * -(y * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.5e-112:
		tmp = t * (x * -y)
	elif y <= 1.06e-12:
		tmp = x
	else:
		tmp = x * -(y * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.5e-112)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.06e-12)
		tmp = x;
	else
		tmp = Float64(x * Float64(-Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.5e-112)
		tmp = t * (x * -y);
	elseif (y <= 1.06e-12)
		tmp = x;
	else
		tmp = x * -(y * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.5e-112], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e-12], x, N[(x * (-N[(y * t), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999992e-112

   1. Initial program 94.8%

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

   2. Taylor expanded in t around inf 53.6%

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

      1. mul-1-neg 53.6%

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

      2. distribute-rgt-neg-out 53.6%

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

   4. Simplified 53.6%

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

   5. Taylor expanded in y around 0 16.7%

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

      1. +-commutative 16.7%

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

      2. mul-1-neg 16.7%

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

      3. unsub-neg 16.7%

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

      4. associate-*r* 17.7%

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

      5. *-commutative 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. expm1-log1p-u 12.6%

         \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(t \cdot x\right)\right)\right)} \]

      2. expm1-udef 20.7%

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

      3. *-commutative 20.7%

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

      4. associate-*l* 22.6%

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

   10. Applied egg-rr 22.6%

       \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 14.6%

          \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)\right)} \]

       2. expm1-log1p 20.5%

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

   12. Simplified 20.5%

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

   if -8.49999999999999992e-112 < y < 1.0599999999999999e-12

   1. Initial program 96.1%

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

   2. Taylor expanded in b around inf 85.4%

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

      1. associate-*r* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in b around 0 39.0%

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

   if 1.0599999999999999e-12 < y

   1. Initial program 95.3%

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

   2. Taylor expanded in t around inf 53.2%

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

      1. mul-1-neg 53.2%

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

      2. distribute-rgt-neg-out 53.2%

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

   4. Simplified 53.2%

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

   5. Taylor expanded in y around 0 11.4%

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

      1. +-commutative 11.4%

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

      2. mul-1-neg 11.4%

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

      3. unsub-neg 11.4%

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

      4. associate-*r* 19.9%

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

      5. *-commutative 19.9%

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

   7. Simplified 19.9%

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

   8. Taylor expanded in y around inf 28.8%

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

      1. mul-1-neg 28.8%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. distribute-rgt-neg-in 25.3%

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

      5. distribute-lft-neg-in 25.3%

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

   10. Simplified 25.3%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-112}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-y \cdot t\right)\\ \end{array} \]

Alternative 13: 27.2% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-112}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7e-112)
   (* t (* x (- y)))
   (if (<= y 1.15e-12) x (* y (* x (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7e-112) {
		tmp = t * (x * -y);
	} else if (y <= 1.15e-12) {
		tmp = x;
	} else {
		tmp = y * (x * -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 <= (-7d-112)) then
        tmp = t * (x * -y)
    else if (y <= 1.15d-12) then
        tmp = x
    else
        tmp = y * (x * -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 <= -7e-112) {
		tmp = t * (x * -y);
	} else if (y <= 1.15e-12) {
		tmp = x;
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7e-112:
		tmp = t * (x * -y)
	elif y <= 1.15e-12:
		tmp = x
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7e-112)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.15e-12)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7e-112)
		tmp = t * (x * -y);
	elseif (y <= 1.15e-12)
		tmp = x;
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7e-112], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e-12], x, N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999988e-112

   1. Initial program 94.8%

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

   2. Taylor expanded in t around inf 53.6%

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

      1. mul-1-neg 53.6%

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

      2. distribute-rgt-neg-out 53.6%

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

   4. Simplified 53.6%

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

   5. Taylor expanded in y around 0 16.7%

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

      1. +-commutative 16.7%

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

      2. mul-1-neg 16.7%

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

      3. unsub-neg 16.7%

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

      4. associate-*r* 17.7%

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

      5. *-commutative 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. expm1-log1p-u 12.6%

         \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(t \cdot x\right)\right)\right)} \]

      2. expm1-udef 20.7%

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

      3. *-commutative 20.7%

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

      4. associate-*l* 22.6%

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

   10. Applied egg-rr 22.6%

       \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 14.6%

          \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(x \cdot y\right)\right)\right)} \]

       2. expm1-log1p 20.5%

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

   12. Simplified 20.5%

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

   if -6.99999999999999988e-112 < y < 1.14999999999999995e-12

   1. Initial program 96.1%

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

   2. Taylor expanded in b around inf 85.4%

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

      1. associate-*r* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in b around 0 39.0%

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

   if 1.14999999999999995e-12 < y

   1. Initial program 95.3%

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

   2. Taylor expanded in t around inf 53.2%

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

      1. mul-1-neg 53.2%

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

      2. distribute-rgt-neg-out 53.2%

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

   4. Simplified 53.2%

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

   5. Taylor expanded in y around 0 11.4%

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

      1. +-commutative 11.4%

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

      2. mul-1-neg 11.4%

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

      3. unsub-neg 11.4%

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

      4. associate-*r* 19.9%

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

      5. *-commutative 19.9%

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

   7. Simplified 19.9%

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

   8. Taylor expanded in y around inf 28.8%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-112}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 14: 23.4% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0033:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= y 0.0033) x (* a (* x b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.0033) {
		tmp = x;
	} else {
		tmp = a * (x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 0.0033d0) then
        tmp = x
    else
        tmp = a * (x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.0033) {
		tmp = x;
	} else {
		tmp = a * (x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.0033:
		tmp = x
	else:
		tmp = a * (x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.0033)
		tmp = x;
	else
		tmp = Float64(a * Float64(x * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.0033)
		tmp = x;
	else
		tmp = a * (x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.0033], x, N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0033

   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. Taylor expanded in b around inf 64.9%

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

      1. associate-*r* 64.9%

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

      2. neg-mul-1 64.9%

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

      3. *-commutative 64.9%

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

   4. Simplified 64.9%

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

   5. Taylor expanded in b around 0 23.0%

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

   if 0.0033 < y

   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. Taylor expanded in b around inf 32.0%

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

      1. associate-*r* 32.0%

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

      2. neg-mul-1 32.0%

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

      3. *-commutative 32.0%

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

   4. Simplified 32.0%

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

   5. Taylor expanded in b around 0 6.5%

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

      1. +-commutative 6.5%

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

      2. mul-1-neg 6.5%

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

      3. unsub-neg 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in a around inf 20.4%

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

      1. neg-mul-1 20.4%

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

      2. distribute-rgt-neg-in 20.4%

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

      3. distribute-lft-neg-in 20.4%

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

   10. Simplified 20.4%

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

       1. expm1-log1p-u 19.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(\left(-b\right) \cdot x\right)\right)\right)} \]

       2. expm1-udef 44.8%

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

       3. add-sqr-sqrt 14.9%

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

       4. sqrt-unprod 43.1%

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

       5. sqr-neg 43.1%

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

       6. sqrt-unprod 28.2%

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

       7. add-sqr-sqrt 41.7%

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

       8. *-commutative 41.7%

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

   12. Applied egg-rr 41.7%

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

       1. expm1-def 16.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(x \cdot b\right)\right)\right)} \]

       2. expm1-log1p 18.7%

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

   14. Simplified 18.7%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0033:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \]

Alternative 15: 19.8% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

2. Taylor expanded in b around inf 57.0%

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

   1. associate-*r* 57.0%

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

   2. neg-mul-1 57.0%

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

   3. *-commutative 57.0%

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

4. Simplified 57.0%

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

5. Taylor expanded in b around 0 18.4%

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

6. Final simplification 18.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023268 
(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))))))