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

Percentage Accurate: 96.6% → 99.2%

Time: 17.9s

Alternatives: 12

Speedup: 1.2×

• 41.3% 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: 96.6% 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

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: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

   8. fp-cancel-sub-sign N/A

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

   9. associate--l- N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. distribute-lft-out N/A

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

   16. lower-*.f64 N/A

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

   17. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 77.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-33}:\\ \;\;\;\;e^{\left(\log \left(1 - z\right) - b\right) \cdot a} \cdot x\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+215}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-a\right) \cdot b} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e-33)
   (* (exp (* (- (log (- 1.0 z)) b) a)) x)
   (if (<= b 1.45e+215)
     (* (exp (* (- (log z) t) y)) x)
     (* (exp (* (- a) b)) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e-33) {
		tmp = exp(((log((1.0 - z)) - b) * a)) * x;
	} else if (b <= 1.45e+215) {
		tmp = exp(((log(z) - t) * y)) * x;
	} else {
		tmp = exp((-a * b)) * 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 (b <= (-5d-33)) then
        tmp = exp(((log((1.0d0 - z)) - b) * a)) * x
    else if (b <= 1.45d+215) then
        tmp = exp(((log(z) - t) * y)) * x
    else
        tmp = exp((-a * b)) * 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 (b <= -5e-33) {
		tmp = Math.exp(((Math.log((1.0 - z)) - b) * a)) * x;
	} else if (b <= 1.45e+215) {
		tmp = Math.exp(((Math.log(z) - t) * y)) * x;
	} else {
		tmp = Math.exp((-a * b)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -5e-33:
		tmp = math.exp(((math.log((1.0 - z)) - b) * a)) * x
	elif b <= 1.45e+215:
		tmp = math.exp(((math.log(z) - t) * y)) * x
	else:
		tmp = math.exp((-a * b)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e-33)
		tmp = Float64(exp(Float64(Float64(log(Float64(1.0 - z)) - b) * a)) * x);
	elseif (b <= 1.45e+215)
		tmp = Float64(exp(Float64(Float64(log(z) - t) * y)) * x);
	else
		tmp = Float64(exp(Float64(Float64(-a) * b)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -5e-33)
		tmp = exp(((log((1.0 - z)) - b) * a)) * x;
	elseif (b <= 1.45e+215)
		tmp = exp(((log(z) - t) * y)) * x;
	else
		tmp = exp((-a * b)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e-33], N[(N[Exp[N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 1.45e+215], N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000028e-33

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 71.3

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

   4. Applied rewrites 71.3%

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

   if -5.00000000000000028e-33 < b < 1.45e215

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 1.45e215 < b

   1. Initial program 96.9%

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

   2. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.6

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

   6. Applied rewrites 85.6%

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

Alternative 3: 77.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+215}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- a) b)) x)))
   (if (<= b -1.05e-14)
     t_1
     (if (<= b 1.45e+215) (* (exp (* (- (log z) t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-a * b)) * x;
	double tmp;
	if (b <= -1.05e-14) {
		tmp = t_1;
	} else if (b <= 1.45e+215) {
		tmp = exp(((log(z) - t) * y)) * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = exp((-a * b)) * x
    if (b <= (-1.05d-14)) then
        tmp = t_1
    else if (b <= 1.45d+215) then
        tmp = exp(((log(z) - t) * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-a * b)) * x;
	double tmp;
	if (b <= -1.05e-14) {
		tmp = t_1;
	} else if (b <= 1.45e+215) {
		tmp = Math.exp(((Math.log(z) - t) * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-a * b)) * x
	tmp = 0
	if b <= -1.05e-14:
		tmp = t_1
	elif b <= 1.45e+215:
		tmp = math.exp(((math.log(z) - t) * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-a) * b)) * x)
	tmp = 0.0
	if (b <= -1.05e-14)
		tmp = t_1;
	elseif (b <= 1.45e+215)
		tmp = Float64(exp(Float64(Float64(log(z) - t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-a * b)) * x;
	tmp = 0.0;
	if (b <= -1.05e-14)
		tmp = t_1;
	elseif (b <= 1.45e+215)
		tmp = exp(((log(z) - t) * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -1.05e-14], t$95$1, If[LessEqual[b, 1.45e+215], N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.0499999999999999e-14 or 1.45e215 < b

   1. Initial program 98.4%

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

   2. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if -1.0499999999999999e-14 < b < 1.45e215

   1. Initial program 95.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 78.9

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

   4. Applied rewrites 78.9%

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

Alternative 4: 69.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-a\right) \cdot b} \cdot x\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 31500000:\\ \;\;\;\;e^{\left(-y\right) \cdot t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- a) b)) x)))
   (if (<= b -4.2e-57)
     t_1
     (if (<= b 31500000.0) (* (exp (* (- y) t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-a * b)) * x;
	double tmp;
	if (b <= -4.2e-57) {
		tmp = t_1;
	} else if (b <= 31500000.0) {
		tmp = exp((-y * t)) * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = exp((-a * b)) * x
    if (b <= (-4.2d-57)) then
        tmp = t_1
    else if (b <= 31500000.0d0) then
        tmp = exp((-y * t)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-a * b)) * x;
	double tmp;
	if (b <= -4.2e-57) {
		tmp = t_1;
	} else if (b <= 31500000.0) {
		tmp = Math.exp((-y * t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-a * b)) * x
	tmp = 0
	if b <= -4.2e-57:
		tmp = t_1
	elif b <= 31500000.0:
		tmp = math.exp((-y * t)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-a) * b)) * x)
	tmp = 0.0
	if (b <= -4.2e-57)
		tmp = t_1;
	elseif (b <= 31500000.0)
		tmp = Float64(exp(Float64(Float64(-y) * t)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-a * b)) * x;
	tmp = 0.0;
	if (b <= -4.2e-57)
		tmp = t_1;
	elseif (b <= 31500000.0)
		tmp = exp((-y * t)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -4.2e-57], t$95$1, If[LessEqual[b, 31500000.0], N[(N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.1999999999999999e-57 or 3.15e7 < b

   1. Initial program 98.5%

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

   2. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.6

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

   6. Applied rewrites 71.6%

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

   if -4.1999999999999999e-57 < b < 3.15e7

   1. Initial program 94.3%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 67.4

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

   4. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.4

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

   6. Applied rewrites 67.4%

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

Alternative 5: 58.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in b around inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 58.4

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

4. Applied rewrites 58.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 58.4

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

6. Applied rewrites 58.4%

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

Alternative 6: 35.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z - t\\ t_2 := y \cdot t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -50000000:\\ \;\;\;\;\left(t \cdot x\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t\_2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (log z) t)) (t_2 (+ (* y t_1) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -50000000.0)
     (* (* t x) (- y))
     (if (<= t_2 0.0005) (fma (* (- t) y) x x) (fma (* t_1 x) y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(z) - t;
	double t_2 = (y * t_1) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = (t * x) * -y;
	} else if (t_2 <= 0.0005) {
		tmp = fma((-t * y), x, x);
	} else {
		tmp = fma((t_1 * x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(z) - t)
	t_2 = Float64(Float64(y * t_1) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -50000000.0)
		tmp = Float64(Float64(t * x) * Float64(-y));
	elseif (t_2 <= 0.0005)
		tmp = fma(Float64(Float64(-t) * y), x, x);
	else
		tmp = fma(Float64(t_1 * x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t$95$1), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -50000000.0], N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$2, 0.0005], N[(N[((-t) * y), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(t$95$1 * x), $MachinePrecision] * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e7

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.7

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

   10. Applied rewrites 17.7%

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

   if -5e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000001e-4

   1. Initial program 92.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if 5.0000000000000001e-4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.6%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

   11. Applied rewrites 28.3%

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

Alternative 7: 34.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;y \cdot t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -50000000:\\ \;\;\;\;\left(t \cdot x\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (log z) t)))
   (if (<= (+ (* y t_1) (* a (- (log (- 1.0 z)) b))) -50000000.0)
     (* (* t x) (- y))
     (fma (* t_1 y) x x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(z) - t;
	double tmp;
	if (((y * t_1) + (a * (log((1.0 - z)) - b))) <= -50000000.0) {
		tmp = (t * x) * -y;
	} else {
		tmp = fma((t_1 * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(z) - t)
	tmp = 0.0
	if (Float64(Float64(y * t_1) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -50000000.0)
		tmp = Float64(Float64(t * x) * Float64(-y));
	else
		tmp = fma(Float64(t_1 * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(N[(y * t$95$1), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -50000000.0], N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(t$95$1 * y), $MachinePrecision] * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e7

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.7

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

   10. Applied rewrites 17.7%

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

   if -5e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

      \[\leadsto \mathsf{fma}\left(\left(\log z - t\right) \cdot y, \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 34.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z - t\\ t_2 := y \cdot t\_1 + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -50000000:\\ \;\;\;\;\left(t \cdot x\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (log z) t)) (t_2 (+ (* y t_1) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -50000000.0)
     (* (* t x) (- y))
     (if (<= t_2 2e+256) (* (fma (log z) y 1.0) x) (* (* t_1 y) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(z) - t;
	double t_2 = (y * t_1) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = (t * x) * -y;
	} else if (t_2 <= 2e+256) {
		tmp = fma(log(z), y, 1.0) * x;
	} else {
		tmp = (t_1 * y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(z) - t)
	t_2 = Float64(Float64(y * t_1) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -50000000.0)
		tmp = Float64(Float64(t * x) * Float64(-y));
	elseif (t_2 <= 2e+256)
		tmp = Float64(fma(log(z), y, 1.0) * x);
	else
		tmp = Float64(Float64(t_1 * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t$95$1), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -50000000.0], N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$2, 2e+256], N[(N[(N[Log[z], $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$1 * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e7

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.7

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

   10. Applied rewrites 17.7%

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

   if -5e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e256

   1. Initial program 95.1%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      4. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      5. lift--.f64 47.4

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   7. Applied rewrites 47.4%

      \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   8. Taylor expanded in t around 0

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

      1. lift-log.f64 44.0

         \[\leadsto \mathsf{fma}\left(\log z, y, 1\right) \cdot x \]

   10. Applied rewrites 44.0%

       \[\leadsto \mathsf{fma}\left(\log z, y, 1\right) \cdot x \]

   if 2.0000000000000001e256 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      4. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      5. lift--.f64 47.7

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   7. Applied rewrites 47.7%

      \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(y \cdot \left(\log z - t\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

       \[\leadsto \left(\left(\log z - t\right) \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 33.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;\left(t \cdot x\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -50000000.0)
     (* (* t x) (- y))
     (if (<= t_1 2e+256) (* (fma (log z) y 1.0) x) (fma (* (- t) y) x x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = (t * x) * -y;
	} else if (t_1 <= 2e+256) {
		tmp = fma(log(z), y, 1.0) * x;
	} else {
		tmp = fma((-t * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = Float64(Float64(t * x) * Float64(-y));
	elseif (t_1 <= 2e+256)
		tmp = Float64(fma(log(z), y, 1.0) * x);
	else
		tmp = fma(Float64(Float64(-t) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$1, 2e+256], N[(N[(N[Log[z], $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e7

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 68.4

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.7

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

   10. Applied rewrites 17.7%

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

   if -5e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e256

   1. Initial program 95.1%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + y \cdot \left(\log z - t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\log z - t\right) + 1\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\log z - t\right) \cdot y + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      4. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

      5. lift--.f64 47.4

         \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   7. Applied rewrites 47.4%

      \[\leadsto \mathsf{fma}\left(\log z - t, y, 1\right) \cdot x \]

   8. Taylor expanded in t around 0

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

      1. lift-log.f64 44.0

         \[\leadsto \mathsf{fma}\left(\log z, y, 1\right) \cdot x \]

   10. Applied rewrites 44.0%

       \[\leadsto \mathsf{fma}\left(\log z, y, 1\right) \cdot x \]

   if 2.0000000000000001e256 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.1

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

   10. Applied rewrites 43.1%

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

Alternative 10: 32.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot x\right) \cdot \left(-y\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -50000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* t x) (- y)))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -50000000.0) t_1 (if (<= t_2 5e+55) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t * x) * -y;
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+55) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * x) * -y
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-50000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+55) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t * x) * -y;
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+55) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t * x) * -y
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -50000000.0:
		tmp = t_1
	elif t_2 <= 5e+55:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t * x) * Float64(-y))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -50000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+55)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t * x) * -y;
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -50000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+55)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -50000000.0], t$95$1, If[LessEqual[t$95$2, 5e+55], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e7 or 5.00000000000000046e55 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 16.9

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 20.7

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

   10. Applied rewrites 20.7%

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

   if -5e7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.00000000000000046e55

   1. Initial program 92.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto x \]
   6. Step-by-step derivation

   7. Applied rewrites 70.3%

      \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 25.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;\left(t \cdot x\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))) 0.0)
   (* (* t x) (- y))
   (fma (* (- t) y) x x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))))) <= 0.0) {
		tmp = (t * x) * -y;
	} else {
		tmp = fma((-t * y), x, x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.0], N[(N[(t * x), $MachinePrecision] * (-y)), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 0.0

   1. Initial program 97.6%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 21.2

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

   7. Applied rewrites 21.2%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.5

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

   10. Applied rewrites 17.5%

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

   if 0.0 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 94.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.7

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

   10. Applied rewrites 44.7%

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

Alternative 12: 19.1% accurate, 42.1× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift--.f64 59.1

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

4. Applied rewrites 59.1%

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

5. Taylor expanded in a around 0

   \[\leadsto x \]
6. Step-by-step derivation

7. Applied rewrites 19.1%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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))))))