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

Percentage Accurate: 96.6% → 99.6%

Time: 11.4s

Alternatives: 10

Speedup: 1.5×

• 40.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

5. Final simplification 98.4%

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

Alternative 2: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-63} \lor \neg \left(b \leq 9 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.1e-63) (not (<= b 9e-82)))
   (* x (exp (fma (- b) a (* (log z) y))))
   (* x (exp (* (- (log z) t) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.1e-63) || !(b <= 9e-82)) {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	} else {
		tmp = x * exp(((log(z) - t) * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.1e-63) || !(b <= 9e-82))
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	else
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.1e-63], N[Not[LessEqual[b, 9e-82]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1e-63 or 8.9999999999999997e-82 < b

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.1%

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

   if -1.1e-63 < b < 8.9999999999999997e-82

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-log.f64 85.6

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

   11. Applied rewrites 85.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-63} \lor \neg \left(b \leq 9 \cdot 10^{-82}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 8.4 \cdot 10^{+21}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -0.375) (not (<= b 8.4e+21)))
   (* x (exp (* (- b) a)))
   (* x (exp (* (- (log z) t) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 8.4e+21)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x * exp(((log(z) - t) * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.375d0)) .or. (.not. (b <= 8.4d+21))) then
        tmp = x * exp((-b * a))
    else
        tmp = x * exp(((log(z) - t) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 8.4e+21)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x * Math.exp(((Math.log(z) - t) * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -0.375) or not (b <= 8.4e+21):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x * math.exp(((math.log(z) - t) * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -0.375) || !(b <= 8.4e+21))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -0.375) || ~((b <= 8.4e+21)))
		tmp = x * exp((-b * a));
	else
		tmp = x * exp(((log(z) - t) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -0.375], N[Not[LessEqual[b, 8.4e+21]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.375 or 8.4e21 < b

   1. Initial program 96.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.2%

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

   if -0.375 < b < 8.4e21

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-log.f64 81.9

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

   11. Applied rewrites 81.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 8.4 \cdot 10^{+21}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 98.4

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

5. Applied rewrites 98.4%

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

Alternative 5: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+99} \lor \neg \left(y \leq 3.5 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot e^{\log z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.9e+99) (not (<= y 3.5e+52)))
   (* x (exp (* (log 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 <= -4.9e+99) || !(y <= 3.5e+52)) {
		tmp = x * exp((log(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 <= (-4.9d+99)) .or. (.not. (y <= 3.5d+52))) then
        tmp = x * exp((log(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 <= -4.9e+99) || !(y <= 3.5e+52)) {
		tmp = x * Math.exp((Math.log(z) * y));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.9e+99) or not (y <= 3.5e+52):
		tmp = x * math.exp((math.log(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 <= -4.9e+99) || !(y <= 3.5e+52))
		tmp = Float64(x * exp(Float64(log(z) * y)));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.9e+99) || ~((y <= 3.5e+52)))
		tmp = x * exp((log(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, -4.9e+99], N[Not[LessEqual[y, 3.5e+52]], $MachinePrecision]], N[(x * N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999997e99 or 3.5e52 < y

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.7%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-log.f64 86.9

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

   11. Applied rewrites 86.9%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 79.2%

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

   if -4.8999999999999997e99 < y < 3.5e52

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+99} \lor \neg \left(y \leq 3.5 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot e^{\log z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 94.1

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

5. Applied rewrites 94.1%

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

Alternative 7: 69.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 2.3 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -0.375) (not (<= b 2.3e-79)))
   (* x (exp (* (- b) a)))
   (/ x (exp (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 2.3e-79)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x / exp((y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.375d0)) .or. (.not. (b <= 2.3d-79))) then
        tmp = x * exp((-b * a))
    else
        tmp = x / exp((y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 2.3e-79)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x / Math.exp((y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -0.375) or not (b <= 2.3e-79):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x / math.exp((y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -0.375) || !(b <= 2.3e-79))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x / exp(Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -0.375) || ~((b <= 2.3e-79)))
		tmp = x * exp((-b * a));
	else
		tmp = x / exp((y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -0.375], N[Not[LessEqual[b, 2.3e-79]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.375 or 2.30000000000000012e-79 < b

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.6%

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

   if -0.375 < b < 2.30000000000000012e-79

   1. Initial program 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot 1}{e^{\color{blue}{y \cdot t}}} \]

      2. lower-*.f64 68.5

         \[\leadsto \frac{x \cdot 1}{e^{\color{blue}{y \cdot t}}} \]

   7. Applied rewrites 68.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{e^{y \cdot t}} \]

      2. *-rgt-identity 68.5

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

   9. Applied rewrites 68.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 2.3 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 2.3 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -0.375) (not (<= b 2.3e-79)))
   (* x (exp (* (- b) a)))
   (* x (exp (* (- y) t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 2.3e-79)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x * exp((-y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.375d0)) .or. (.not. (b <= 2.3d-79))) then
        tmp = x * exp((-b * a))
    else
        tmp = x * exp((-y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -0.375) || !(b <= 2.3e-79)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x * Math.exp((-y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -0.375) or not (b <= 2.3e-79):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x * math.exp((-y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -0.375) || !(b <= 2.3e-79))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -0.375) || ~((b <= 2.3e-79)))
		tmp = x * exp((-b * a));
	else
		tmp = x * exp((-y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -0.375], N[Not[LessEqual[b, 2.3e-79]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.375 or 2.30000000000000012e-79 < b

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.6%

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

   if -0.375 < b < 2.30000000000000012e-79

   1. Initial program 92.9%

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

   3. Taylor expanded in 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-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 68.5

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

   5. Applied rewrites 68.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.375 \lor \neg \left(b \leq 2.3 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-131} \lor \neg \left(b \leq 3.8 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -4.5e-131) (not (<= b 3.8e-90)))
   (* x (exp (* (- b) a)))
   (* x (exp (* (- a) z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -4.5e-131) || !(b <= 3.8e-90)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x * exp((-a * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.5d-131)) .or. (.not. (b <= 3.8d-90))) then
        tmp = x * exp((-b * a))
    else
        tmp = x * exp((-a * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -4.5e-131) || !(b <= 3.8e-90)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x * Math.exp((-a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -4.5e-131) or not (b <= 3.8e-90):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x * math.exp((-a * z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -4.5e-131) || !(b <= 3.8e-90))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -4.5e-131) || ~((b <= 3.8e-90)))
		tmp = x * exp((-b * a));
	else
		tmp = x * exp((-a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -4.5e-131], N[Not[LessEqual[b, 3.8e-90]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5000000000000002e-131 or 3.8e-90 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.5%

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

   if -4.5000000000000002e-131 < b < 3.8e-90

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-131} \lor \neg \left(b \leq 3.8 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 34.5%

   \[\leadsto x \cdot e^{\left(-a\right) \cdot \color{blue}{z}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))))