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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

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, c, i)
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), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

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, c, i)
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), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, a\right)\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (fma (log y) x (+ (+ t z) (fma (log c) (- b 0.5) a))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(log(y), x, ((t + z) + fma(log(c), (b - 0.5), a))) + (y * i);
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(fma(log(y), x, Float64(Float64(t + z) + fma(log(c), Float64(b - 0.5), a))) + Float64(y * i))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   5. lift-+.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   6. associate-+l+ N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \log y + \left(z + t\right)\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   7. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right)} + y \cdot i \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \log y} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\log y \cdot x} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

   11. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   13. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\log c \cdot \left(b - \frac{1}{2}\right)} + a\right)\right) + y \cdot i \]

   17. lower-fma.f64 99.9

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - 0.5, a\right)}\right) + y \cdot i \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)} + y \cdot i \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (fma y i (fma x (log y) (+ (fma (- b 0.5) (log c) z) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(y, i, fma(x, log(y), (fma((b - 0.5), log(c), z) + a)));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(y, i, fma(x, log(y), Float64(fma(Float64(b - 0.5), log(c), z) + a)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(x * N[Log[y], $MachinePrecision] + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

5. Applied rewrites 87.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]
6. Step-by-step derivation

7. Applied rewrites 87.5%

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{i}, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\right) \]

8. Final simplification 87.5%

   \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\right) \]
9. Add Preprocessing

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (fma i y (fma (log y) x (fma (- b 0.5) (log c) z))) a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(i, y, fma(log(y), x, fma((b - 0.5), log(c), z))) + a;
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(fma(i, y, fma(log(y), x, fma(Float64(b - 0.5), log(c), z))) + a)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

5. Applied rewrites 87.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

6. Final simplification 87.5%

   \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a \]
7. Add Preprocessing

Alternative 4: 91.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+99} \lor \neg \left(x \leq 1.4 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, 1 \cdot z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.4e+99) (not (<= x 1.4e+84)))
   (+ (fma i y (fma (log y) x (* 1.0 z))) a)
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.4e+99) || !(x <= 1.4e+84)) {
		tmp = fma(i, y, fma(log(y), x, (1.0 * z))) + a;
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.4e+99) || !(x <= 1.4e+84))
		tmp = Float64(fma(i, y, fma(log(y), x, Float64(1.0 * z))) + a);
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.4e+99], N[Not[LessEqual[x, 1.4e+84]], $MachinePrecision]], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e99 or 1.39999999999999991e84 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right)\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 87.5%

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z\right)\right) + a \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 87.1%

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

   if -1.4e99 < x < 1.39999999999999991e84

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\mathsf{fma}\left(b - \frac{1}{2}, \log c, z + i \cdot y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \left(a + t\right) + \mathsf{fma}\left(\color{blue}{b - \frac{1}{2}}, \log c, z + i \cdot y\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - \frac{1}{2}, \color{blue}{\log c}, z + i \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - \frac{1}{2}, \log c, \color{blue}{i \cdot y + z}\right) \]

      11. lower-fma.f64 100.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+99} \lor \neg \left(x \leq 1.4 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, 1 \cdot z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+99} \lor \neg \left(x \leq 1.4 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, 1 \cdot z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.4e+99) (not (<= x 1.4e+84)))
   (+ (fma i y (fma (log y) x (* 1.0 z))) a)
   (+ (fma i y (fma (log c) (- b 0.5) z)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.4e+99) || !(x <= 1.4e+84)) {
		tmp = fma(i, y, fma(log(y), x, (1.0 * z))) + a;
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), z)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.4e+99) || !(x <= 1.4e+84))
		tmp = Float64(fma(i, y, fma(log(y), x, Float64(1.0 * z))) + a);
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), z)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.4e+99], N[Not[LessEqual[x, 1.4e+84]], $MachinePrecision]], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e99 or 1.39999999999999991e84 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right)\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 87.5%

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z\right)\right) + a \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 87.1%

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

   if -1.4e99 < x < 1.39999999999999991e84

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(i, y, z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 84.8%

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+99} \lor \neg \left(x \leq 1.4 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, 1 \cdot z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 1.36 \cdot 10^{+244}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.05e+202) (not (<= x 1.36e+244)))
   (* (log y) x)
   (+ (fma i y (fma (log c) (- b 0.5) z)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.05e+202) || !(x <= 1.36e+244)) {
		tmp = log(y) * x;
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), z)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.05e+202) || !(x <= 1.36e+244))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), z)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.05e+202], N[Not[LessEqual[x, 1.36e+244]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e202 or 1.36e244 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      6. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \log y + \left(z + t\right)\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right)} + y \cdot i \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log y \cdot x} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\log c \cdot \left(b - \frac{1}{2}\right)} + a\right)\right) + y \cdot i \]

      17. lower-fma.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - 0.5, a\right)}\right) + y \cdot i \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)} + y \cdot i \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -1.05e202 < x < 1.36e244

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(i, y, z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 1.36 \cdot 10^{+244}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 2.6 \cdot 10^{+240}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(-0.5, \log c, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.05e+202) (not (<= x 2.6e+240)))
   (* (log y) x)
   (+ (+ a t) (fma i y (fma -0.5 (log c) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.05e+202) || !(x <= 2.6e+240)) {
		tmp = log(y) * x;
	} else {
		tmp = (a + t) + fma(i, y, fma(-0.5, log(c), z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.05e+202) || !(x <= 2.6e+240))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(a + t) + fma(i, y, fma(-0.5, log(c), z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.05e+202], N[Not[LessEqual[x, 2.6e+240]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(i * y + N[(-0.5 * N[Log[c], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e202 or 2.6e240 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      6. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \log y + \left(z + t\right)\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right)} + y \cdot i \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log y \cdot x} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\log c \cdot \left(b - \frac{1}{2}\right)} + a\right)\right) + y \cdot i \]

      17. lower-fma.f64 99.7

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - 0.5, a\right)}\right) + y \cdot i \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)} + y \cdot i \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -1.05e202 < x < 2.6e240

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\mathsf{fma}\left(b - \frac{1}{2}, \log c, z + i \cdot y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \left(a + t\right) + \mathsf{fma}\left(\color{blue}{b - \frac{1}{2}}, \log c, z + i \cdot y\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - \frac{1}{2}, \color{blue}{\log c}, z + i \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - \frac{1}{2}, \log c, \color{blue}{i \cdot y + z}\right) \]

      11. lower-fma.f64 94.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \left(a + t\right) + \left(z + \color{blue}{\left(\frac{-1}{2} \cdot \log c + i \cdot y\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.6%

      \[\leadsto \left(a + t\right) + \mathsf{fma}\left(i, \color{blue}{y}, \mathsf{fma}\left(-0.5, \log c, z\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 2.6 \cdot 10^{+240}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(-0.5, \log c, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.8 \cdot 10^{-33} \lor \neg \left(i \leq 12.5\right):\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= i -3.8e-33) (not (<= i 12.5)))
   (+ (* (+ (/ z i) y) i) a)
   (+ (fma (- b 0.5) (log c) z) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i <= -3.8e-33) || !(i <= 12.5)) {
		tmp = (((z / i) + y) * i) + a;
	} else {
		tmp = fma((b - 0.5), log(c), z) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((i <= -3.8e-33) || !(i <= 12.5))
		tmp = Float64(Float64(Float64(Float64(z / i) + y) * i) + a);
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), z) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[i, -3.8e-33], N[Not[LessEqual[i, 12.5]], $MachinePrecision]], N[(N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -3.79999999999999994e-33 or 12.5 < i

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in i around inf

      \[\leadsto i \cdot \left(y + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 89.1%

      \[\leadsto \left(\frac{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)}{i} + y\right) \cdot i + a \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]
   10. Step-by-step derivation

   11. Applied rewrites 65.7%

       \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]

   if -3.79999999999999994e-33 < i < 12.5

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in y around 0

      \[\leadsto a + \color{blue}{\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.7%

      \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + \color{blue}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, a + z\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto a + \left(z + \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 59.8%

       \[\leadsto \mathsf{fma}\left(b - 0.5, \log c, z\right) + a \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.8 \cdot 10^{-33} \lor \neg \left(i \leq 12.5\right):\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 2.4 \cdot 10^{+240}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.05e+202) (not (<= x 2.4e+240)))
   (* (log y) x)
   (+ (* (+ (/ z i) y) i) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.05e+202) || !(x <= 2.4e+240)) {
		tmp = log(y) * x;
	} else {
		tmp = (((z / i) + y) * i) + a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((x <= (-1.05d+202)) .or. (.not. (x <= 2.4d+240))) then
        tmp = log(y) * x
    else
        tmp = (((z / i) + y) * i) + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.05e+202) || !(x <= 2.4e+240)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (((z / i) + y) * i) + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -1.05e+202) or not (x <= 2.4e+240):
		tmp = math.log(y) * x
	else:
		tmp = (((z / i) + y) * i) + a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.05e+202) || !(x <= 2.4e+240))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(z / i) + y) * i) + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -1.05e+202) || ~((x <= 2.4e+240)))
		tmp = log(y) * x;
	else
		tmp = (((z / i) + y) * i) + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.05e+202], N[Not[LessEqual[x, 2.4e+240]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e202 or 2.3999999999999999e240 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      6. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \log y + \left(z + t\right)\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right)} + y \cdot i \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log y \cdot x} + \left(\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)\right) + y \cdot i \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right)} + y \cdot i \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z + t\right) + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)}\right) + y \cdot i \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(t + z\right)} + \left(a + \left(b - \frac{1}{2}\right) \cdot \log c\right)\right) + y \cdot i \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\left(\left(b - \frac{1}{2}\right) \cdot \log c + a\right)}\right) + y \cdot i \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + a\right)\right) + y \cdot i \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \left(\color{blue}{\log c \cdot \left(b - \frac{1}{2}\right)} + a\right)\right) + y \cdot i \]

      17. lower-fma.f64 99.7

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(t + z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - 0.5, a\right)}\right) + y \cdot i \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)} + y \cdot i \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -1.05e202 < x < 2.3999999999999999e240

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

   6. Taylor expanded in i around inf

      \[\leadsto i \cdot \left(y + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 70.1%

      \[\leadsto \left(\frac{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)}{i} + y\right) \cdot i + a \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]
   10. Step-by-step derivation

   11. Applied rewrites 54.0%

       \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+202} \lor \neg \left(x \leq 2.4 \cdot 10^{+240}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{i} + y\right) \cdot i + a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{i} + y\right) \cdot i + a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (+ (* (+ (/ z i) y) i) a))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
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), intent (in) :: c
    real(8), intent (in) :: i
    code = (((z / i) + y) * i) + a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((z / i) + y) * i) + a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((z / i) + y) * i) + a

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(z / i), $MachinePrecision] + y), $MachinePrecision] * i), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

5. Applied rewrites 87.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, z\right)\right)\right) + a} \]

6. Taylor expanded in i around inf

   \[\leadsto i \cdot \left(y + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right) + a \]
7. Step-by-step derivation

8. Applied rewrites 67.6%

   \[\leadsto \left(\frac{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)}{i} + y\right) \cdot i + a \]

9. Taylor expanded in z around inf

   \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]
10. Step-by-step derivation

11. Applied rewrites 46.8%

    \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]

12. Final simplification 46.8%

    \[\leadsto \left(\frac{z}{i} + y\right) \cdot i + a \]
13. Add Preprocessing

Alternative 11: 23.8% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ i \cdot y \end{array} \]

FPCore

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

C

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

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, c, i)
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), intent (in) :: c
    real(8), intent (in) :: i
    code = i * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(i * y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 25.1

      \[\leadsto \color{blue}{i \cdot y} \]

5. Applied rewrites 25.1%

   \[\leadsto \color{blue}{i \cdot y} \]

6. Final simplification 25.1%

   \[\leadsto i \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))