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

Percentage Accurate: 96.4% → 99.6%

Time: 17.8s

Alternatives: 11

Speedup: 1.5×

• Could not converge on a ground truth

  1. x = 1.762178110240275e-195
  2. y = -1.4952388667110849e-296
  3. z = 1.888308139548208e-209
  4. t = -4.992908705696257e+117
  5. a = -4.085832638100806e+216
  6. b = 1.4497933098403665e-260

• 41.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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, 1.5× 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 97.8%

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

3. Taylor expanded in z around 0

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

Alternative 2: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+166} \lor \neg \left(y \leq 2.8 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5e+166) (not (<= y 2.8e+78)))
   (* x (exp (* (- (log z) t) y)))
   (* (exp (* (+ (- y) (/ (* (- b) a) t)) t)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e+166) || !(y <= 2.8e+78)) {
		tmp = x * exp(((log(z) - t) * y));
	} else {
		tmp = exp(((-y + ((-b * a) / t)) * t)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5d+166)) .or. (.not. (y <= 2.8d+78))) then
        tmp = x * exp(((log(z) - t) * y))
    else
        tmp = exp(((-y + ((-b * a) / t)) * t)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e+166) || !(y <= 2.8e+78)) {
		tmp = x * Math.exp(((Math.log(z) - t) * y));
	} else {
		tmp = Math.exp(((-y + ((-b * a) / t)) * t)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5e+166) or not (y <= 2.8e+78):
		tmp = x * math.exp(((math.log(z) - t) * y))
	else:
		tmp = math.exp(((-y + ((-b * a) / t)) * t)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5e+166) || !(y <= 2.8e+78))
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	else
		tmp = Float64(exp(Float64(Float64(Float64(-y) + Float64(Float64(Float64(-b) * a) / t)) * t)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5e+166) || ~((y <= 2.8e+78)))
		tmp = x * exp(((log(z) - t) * y));
	else
		tmp = exp(((-y + ((-b * a) / t)) * t)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5e+166], N[Not[LessEqual[y, 2.8e+78]], $MachinePrecision]], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[((-y) + N[(N[((-b) * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000002e166 or 2.8000000000000001e78 < 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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.8%

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

   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 96.2

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

   11. Applied rewrites 96.2%

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

   if -5.0000000000000002e166 < y < 2.8000000000000001e78

   1. Initial program 96.9%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.2

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

   10. Applied rewrites 94.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+166} \lor \neg \left(y \leq 2.8 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -105000000:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -105000000.0)
   (* (exp (* (+ (- y) (/ (* (- b) a) t)) t)) x)
   (* x (exp (fma (- b) a (* (log z) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -105000000.0) {
		tmp = exp(((-y + ((-b * a) / t)) * t)) * x;
	} else {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -105000000.0)
		tmp = Float64(exp(Float64(Float64(Float64(-y) + Float64(Float64(Float64(-b) * a) / t)) * t)) * x);
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -105000000.0], N[(N[Exp[N[(N[((-y) + N[(N[((-b) * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e8

   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 t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 93.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.9

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

   10. Applied rewrites 96.4%

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

   if -1.05e8 < t

   1. Initial program 98.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   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 95.2%

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -105000000:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% 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 97.8%

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

3. Taylor expanded in z around 0

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
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 97.1

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

5. Applied rewrites 97.1%

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

Alternative 5: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.12 \cdot 10^{+228}:\\ \;\;\;\;x \cdot e^{y \cdot \log z}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.12e+228)
   (* x (exp (* y (log z))))
   (* (exp (* (+ (- y) (/ (* (- b) a) t)) t)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.12e+228) {
		tmp = x * exp((y * log(z)));
	} else {
		tmp = exp(((-y + ((-b * a) / t)) * t)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.12d+228)) then
        tmp = x * exp((y * log(z)))
    else
        tmp = exp(((-y + ((-b * a) / t)) * t)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.12e+228) {
		tmp = x * Math.exp((y * Math.log(z)));
	} else {
		tmp = Math.exp(((-y + ((-b * a) / t)) * t)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.12e+228:
		tmp = x * math.exp((y * math.log(z)))
	else:
		tmp = math.exp(((-y + ((-b * a) / t)) * t)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.12e+228)
		tmp = Float64(x * exp(Float64(y * log(z))));
	else
		tmp = Float64(exp(Float64(Float64(Float64(-y) + Float64(Float64(Float64(-b) * a) / t)) * t)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.12e+228)
		tmp = x * exp((y * log(z)));
	else
		tmp = exp(((-y + ((-b * a) / t)) * t)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.12e+228], N[(x * N[Exp[N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[((-y) + N[(N[((-b) * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.12000000000000003e228

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.8%

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

   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 93.0

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

   11. Applied rewrites 93.0%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 93.0%

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

   if -2.12000000000000003e228 < y

   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 t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.7

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

   10. Applied rewrites 87.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.12 \cdot 10^{+228}:\\ \;\;\;\;x \cdot e^{y \cdot \log z}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(-y\right) + \frac{\left(-b\right) \cdot a}{t}\right) \cdot t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-183} \lor \neg \left(y \leq 3.9 \cdot 10^{-200}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot \left(y - \left(-a\right) \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4e-183) (not (<= y 3.9e-200)))
   (* x (exp (* (- t) (- y (* (- a) (/ b t))))))
   (* x (exp (* (- (- z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4e-183) || !(y <= 3.9e-200)) {
		tmp = x * exp((-t * (y - (-a * (b / t)))));
	} else {
		tmp = x * exp(((-z - b) * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-4d-183)) .or. (.not. (y <= 3.9d-200))) then
        tmp = x * exp((-t * (y - (-a * (b / t)))))
    else
        tmp = x * exp(((-z - 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 <= -4e-183) || !(y <= 3.9e-200)) {
		tmp = x * Math.exp((-t * (y - (-a * (b / t)))));
	} else {
		tmp = x * Math.exp(((-z - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4e-183) or not (y <= 3.9e-200):
		tmp = x * math.exp((-t * (y - (-a * (b / t)))))
	else:
		tmp = x * math.exp(((-z - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4e-183) || !(y <= 3.9e-200))
		tmp = Float64(x * exp(Float64(Float64(-t) * Float64(y - Float64(Float64(-a) * Float64(b / t))))));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4e-183) || ~((y <= 3.9e-200)))
		tmp = x * exp((-t * (y - (-a * (b / t)))));
	else
		tmp = x * exp(((-z - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4e-183], N[Not[LessEqual[y, 3.9e-200]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * N[(y - N[((-a) * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.00000000000000002e-183 or 3.89999999999999999e-200 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 84.3%

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

   if -4.00000000000000002e-183 < y < 3.89999999999999999e-200

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-183} \lor \neg \left(y \leq 3.9 \cdot 10^{-200}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot \left(y - \left(-a\right) \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 = exp(((-y + ((-b * a) / t)) * t)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in t around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. div-add-rev N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. div-add-rev N/A

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

   11. lower-/.f64 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 83.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 83.6

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

10. Applied rewrites 85.1%

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

11. Final simplification 85.1%

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

Alternative 8: 74.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+38} \lor \neg \left(t \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -5.8e+38) (not (<= t 1.9e+63)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- (- z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -5.8e+38) || !(t <= 1.9e+63)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp(((-z - b) * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-5.8d+38)) .or. (.not. (t <= 1.9d+63))) then
        tmp = x * exp((-y * t))
    else
        tmp = x * exp(((-z - 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 ((t <= -5.8e+38) || !(t <= 1.9e+63)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp(((-z - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -5.8e+38) or not (t <= 1.9e+63):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp(((-z - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -5.8e+38) || !(t <= 1.9e+63))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -5.8e+38) || ~((t <= 1.9e+63)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp(((-z - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -5.8e+38], N[Not[LessEqual[t, 1.9e+63]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.80000000000000013e38 or 1.9000000000000001e63 < t

   1. Initial program 98.1%

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

   3. Taylor expanded in 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 84.8

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

   5. Applied rewrites 84.8%

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

   if -5.80000000000000013e38 < t < 1.9000000000000001e63

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+38} \lor \neg \left(t \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+38} \lor \neg \left(t \leq 2.05 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -4.2e+38) (not (<= t 2.05e+29)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.2e+38) || !(t <= 2.05e+29)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp((-b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-4.2d+38)) .or. (.not. (t <= 2.05d+29))) then
        tmp = x * exp((-y * t))
    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 ((t <= -4.2e+38) || !(t <= 2.05e+29)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -4.2e+38) or not (t <= 2.05e+29):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -4.2e+38) || !(t <= 2.05e+29))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	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 ((t <= -4.2e+38) || ~((t <= 2.05e+29)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -4.2e+38], N[Not[LessEqual[t, 2.05e+29]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2e38 or 2.0500000000000002e29 < t

   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 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 82.3

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

   5. Applied rewrites 82.3%

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

   if -4.2e38 < t < 2.0500000000000002e29

   1. Initial program 98.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

   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 73.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+38} \lor \neg \left(t \leq 2.05 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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((-b * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in z around 0

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + y \cdot \left(\log z - t\right)}} \]
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 97.1

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

5. Applied rewrites 97.1%

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

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 60.3%

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

Alternative 11: 33.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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((z * -a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in z around 0

   \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + \left(-1 \cdot \left(a \cdot z\right) + y \cdot \left(\log z - t\right)\right)}} \]
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 33.6%

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

9. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2025016 
(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))))))