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

Percentage Accurate: 96.3% → 96.3%

Time: 12.7s

Alternatives: 10

Speedup: 1.0×

• 40.7% 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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% 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 97.3%

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

Alternative 2: 34.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-t\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -500.0)
     (* y (* t (- x)))
     (if (<= t_1 2e-83) (fma (* y (- t)) x x) (* t (- (/ x t) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = y * (t * -x);
	} else if (t_1 <= 2e-83) {
		tmp = fma((y * -t), x, x);
	} else {
		tmp = t * ((x / t) - (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = Float64(y * Float64(t * Float64(-x)));
	elseif (t_1 <= 2e-83)
		tmp = fma(Float64(y * Float64(-t)), x, x);
	else
		tmp = Float64(t * Float64(Float64(x / t) - Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -500.0], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-83], N[(N[(y * (-t)), $MachinePrecision] * x + x), $MachinePrecision], N[(t * N[(N[(x / t), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -500

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 19.4%

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

   13. Applied rewrites 21.3%

       \[\leadsto \left(t \cdot x\right) \cdot \left(-y\right) \]

   if -500 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-83

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.1%

      \[\leadsto \mathsf{fma}\left(x \cdot y, \color{blue}{\log z - t}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 92.2%

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

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(-1 \cdot \left(t \cdot y\right), x, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 92.2%

       \[\leadsto \mathsf{fma}\left(t \cdot \left(-y\right), x, x\right) \]

   if 2.0000000000000001e-83 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in t around -inf

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

   11. Applied rewrites 34.2%

       \[\leadsto \left(x \cdot y - \frac{\mathsf{fma}\left(\log z, x \cdot y, x\right)}{t}\right) \cdot \left(-t\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(x \cdot y - \frac{x}{t}\right) \cdot \left(\mathsf{neg}\left(t\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 37.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -500:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-t\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -500:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-t\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -500.0)
   (* y (* t (- x)))
   (fma (* y (- t)) x x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -500.0) {
		tmp = y * (t * -x);
	} else {
		tmp = fma((y * -t), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -500.0)
		tmp = Float64(y * Float64(t * Float64(-x)));
	else
		tmp = fma(Float64(y * Float64(-t)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -500.0], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-t)), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -500

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 19.4%

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

   13. Applied rewrites 21.3%

       \[\leadsto \left(t \cdot x\right) \cdot \left(-y\right) \]

   if -500 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 45.0%

      \[\leadsto \mathsf{fma}\left(x \cdot y, \color{blue}{\log z - t}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

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

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(-1 \cdot \left(t \cdot y\right), x, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 44.3%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -500:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-t\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 850000000000:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -5.2e+30)
     t_1
     (if (<= y 850000000000.0) (* x (exp (* a (- (- z) b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -5.2e+30) {
		tmp = t_1;
	} else if (y <= 850000000000.0) {
		tmp = x * exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-5.2d+30)) then
        tmp = t_1
    else if (y <= 850000000000.0d0) then
        tmp = x * exp((a * (-z - b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -5.2e+30) {
		tmp = t_1;
	} else if (y <= 850000000000.0) {
		tmp = x * Math.exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -5.2e+30:
		tmp = t_1
	elif y <= 850000000000.0:
		tmp = x * math.exp((a * (-z - b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -5.2e+30)
		tmp = t_1;
	elseif (y <= 850000000000.0)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -5.2e+30)
		tmp = t_1;
	elseif (y <= 850000000000.0)
		tmp = x * exp((a * (-z - b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+30], t$95$1, If[LessEqual[y, 850000000000.0], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999977e30 or 8.5e11 < 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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -5.19999999999999977e30 < y < 8.5e11

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 87.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 850000000000:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \log z}\\ \mathbf{if}\;y \leq -1.56 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (log z))))))
   (if (<= y -1.56e+27)
     t_1
     (if (<= y 2.1e+14) (* x (exp (* a (- (- z) b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * log(z)));
	double tmp;
	if (y <= -1.56e+27) {
		tmp = t_1;
	} else if (y <= 2.1e+14) {
		tmp = x * exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * log(z)))
    if (y <= (-1.56d+27)) then
        tmp = t_1
    else if (y <= 2.1d+14) then
        tmp = x * exp((a * (-z - b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * Math.log(z)));
	double tmp;
	if (y <= -1.56e+27) {
		tmp = t_1;
	} else if (y <= 2.1e+14) {
		tmp = x * Math.exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * math.log(z)))
	tmp = 0
	if y <= -1.56e+27:
		tmp = t_1
	elif y <= 2.1e+14:
		tmp = x * math.exp((a * (-z - b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * log(z))))
	tmp = 0.0
	if (y <= -1.56e+27)
		tmp = t_1;
	elseif (y <= 2.1e+14)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * log(z)));
	tmp = 0.0;
	if (y <= -1.56e+27)
		tmp = t_1;
	elseif (y <= 2.1e+14)
		tmp = x * exp((a * (-z - b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.56e+27], t$95$1, If[LessEqual[y, 2.1e+14], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.56e27 or 2.1e14 < 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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 72.6%

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

   if -1.56e27 < y < 2.1e14

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 87.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+27}:\\ \;\;\;\;x \cdot e^{y \cdot \log z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+101}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= y -1.55e+67)
     t_1
     (if (<= y 8.5e+101) (* x (exp (* a (- (- z) b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (y <= -1.55e+67) {
		tmp = t_1;
	} else if (y <= 8.5e+101) {
		tmp = x * exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * -t))
    if (y <= (-1.55d+67)) then
        tmp = t_1
    else if (y <= 8.5d+101) then
        tmp = x * exp((a * (-z - b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (y <= -1.55e+67) {
		tmp = t_1;
	} else if (y <= 8.5e+101) {
		tmp = x * Math.exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if y <= -1.55e+67:
		tmp = t_1
	elif y <= 8.5e+101:
		tmp = x * math.exp((a * (-z - b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (y <= -1.55e+67)
		tmp = t_1;
	elseif (y <= 8.5e+101)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (y <= -1.55e+67)
		tmp = t_1;
	elseif (y <= 8.5e+101)
		tmp = x * exp((a * (-z - b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e+67], t$95$1, If[LessEqual[y, 8.5e+101], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.54999999999999998e67 or 8.5000000000000001e101 < 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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.54999999999999998e67 < y < 8.5000000000000001e101

   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 y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+101}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))))
   (if (<= b -1.3e+103) t_1 (if (<= b 2.6e-29) (* x (exp (* y (- t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double tmp;
	if (b <= -1.3e+103) {
		tmp = t_1;
	} else if (b <= 2.6e-29) {
		tmp = x * exp((y * -t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    if (b <= (-1.3d+103)) then
        tmp = t_1
    else if (b <= 2.6d-29) then
        tmp = x * exp((y * -t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((a * -b));
	double tmp;
	if (b <= -1.3e+103) {
		tmp = t_1;
	} else if (b <= 2.6e-29) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	tmp = 0
	if b <= -1.3e+103:
		tmp = t_1
	elif b <= 2.6e-29:
		tmp = x * math.exp((y * -t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	tmp = 0.0
	if (b <= -1.3e+103)
		tmp = t_1;
	elseif (b <= 2.6e-29)
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	tmp = 0.0;
	if (b <= -1.3e+103)
		tmp = t_1;
	elseif (b <= 2.6e-29)
		tmp = x * exp((y * -t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.3e+103], t$95$1, If[LessEqual[b, 2.6e-29], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3000000000000001e103 or 2.6000000000000002e-29 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -1.3000000000000001e103 < b < 2.6000000000000002e-29

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+103}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp((a * -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((a * -b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 58.1

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

5. Applied rewrites 58.1%

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

6. Final simplification 58.1%

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

Alternative 9: 17.4% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(t \cdot \left(-x\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 58.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 18.8%

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

13. Applied rewrites 19.1%

    \[\leadsto \left(t \cdot x\right) \cdot \left(-y\right) \]

14. Final simplification 19.1%

    \[\leadsto y \cdot \left(t \cdot \left(-x\right)\right) \]
15. Add Preprocessing

Alternative 10: 16.8% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(y \cdot \left(-x\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 58.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 28.8%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 18.8%

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

12. Final simplification 18.8%

    \[\leadsto t \cdot \left(y \cdot \left(-x\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))))