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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 21

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. associate-+r+ N/A

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

   12. associate-+r+ N/A

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. 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. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+r+ N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 4: 90.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 85.1

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

4. Applied rewrites 85.1%

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

Alternative 5: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -2.95e+93)
   (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)
   (+ a (fma i y (fma x (log y) (* (log c) (- b 0.5)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2.95e+93) {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	} else {
		tmp = a + fma(i, y, fma(x, log(y), (log(c) * (b - 0.5))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2.95e+93)
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	else
		tmp = Float64(a + fma(i, y, fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2.95e+93], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], N[(a + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.95000000000000004e93

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

   if -2.95000000000000004e93 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 70.5

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

   7. Applied rewrites 70.5%

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

Alternative 6: 88.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \mathsf{fma}\left(\log y, x, t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 6.4e+15)
   (fma (- b 0.5) (log c) (+ a (fma (log y) x (+ t z))))
   (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 6.4e+15) {
		tmp = fma((b - 0.5), log(c), (a + fma(log(y), x, (t + z))));
	} else {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 6.4e+15)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + fma(log(y), x, Float64(t + z))));
	else
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 6.4e+15], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(N[Log[y], $MachinePrecision] * x + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e15

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 76.7

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

   8. Applied rewrites 76.7%

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

   if 6.4e15 < y

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

Alternative 7: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 6.4e+15)
   (+ a (+ z (fma x (log y) (* (log c) (- b 0.5)))))
   (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 6.4e+15) {
		tmp = a + (z + fma(x, log(y), (log(c) * (b - 0.5))));
	} else {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 6.4e+15)
		tmp = Float64(a + Float64(z + fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 6.4e+15], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e15

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 62.1

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

   7. Applied rewrites 62.1%

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

   if 6.4e15 < y

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

Alternative 8: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, a + \mathsf{fma}\left(\log y, x, t + z\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -1.55e+211)
   (fma -0.5 (log c) (+ a (fma (log y) x (+ t z))))
   (if (<= x 3.5e+202)
     (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)
     (fma (/ (* (log y) x) a) a (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -1.55e+211) {
		tmp = fma(-0.5, log(c), (a + fma(log(y), x, (t + z))));
	} else if (x <= 3.5e+202) {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	} else {
		tmp = fma(((log(y) * x) / a), a, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -1.55e+211)
		tmp = fma(-0.5, log(c), Float64(a + fma(log(y), x, Float64(t + z))));
	elseif (x <= 3.5e+202)
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	else
		tmp = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -1.55e+211], N[(-0.5 * N[Log[c], $MachinePrecision] + N[(a + N[(N[Log[y], $MachinePrecision] * x + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+202], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5500000000000001e211

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2}}, \log c, a + \mathsf{fma}\left(\log y, x, t + z\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 61.0%

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

   if -1.5500000000000001e211 < x < 3.49999999999999987e202

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

   if 3.49999999999999987e202 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

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

Alternative 9: 85.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \mathsf{fma}\left(\log y, x, t\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -6.2e+218)
   (fma (- b 0.5) (log c) (+ a (fma (log y) x t)))
   (if (<= x 3.5e+202)
     (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)
     (fma (/ (* (log y) x) a) a (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -6.2e+218) {
		tmp = fma((b - 0.5), log(c), (a + fma(log(y), x, t)));
	} else if (x <= 3.5e+202) {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	} else {
		tmp = fma(((log(y) * x) / a), a, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -6.2e+218)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + fma(log(y), x, t)));
	elseif (x <= 3.5e+202)
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	else
		tmp = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -6.2e+218], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(N[Log[y], $MachinePrecision] * x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+202], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2000000000000003e218

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 62.0%

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

   if -6.2000000000000003e218 < x < 3.49999999999999987e202

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

   if 3.49999999999999987e202 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

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

Alternative 10: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* (log y) x) a) a (* y i))))
   (if (<= x -3.05e+212)
     t_1
     (if (<= x 3.5e+202)
       (+ (fma (log c) (- b 0.5) (fma y i (+ z a))) t)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(((log(y) * x) / a), a, (y * i));
	double tmp;
	if (x <= -3.05e+212) {
		tmp = t_1;
	} else if (x <= 3.5e+202) {
		tmp = fma(log(c), (b - 0.5), fma(y, i, (z + a))) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i))
	tmp = 0.0
	if (x <= -3.05e+212)
		tmp = t_1;
	elseif (x <= 3.5e+202)
		tmp = Float64(fma(log(c), Float64(b - 0.5), fma(y, i, Float64(z + a))) + t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.05e+212], t$95$1, If[LessEqual[x, 3.5e+202], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0499999999999999e212 or 3.49999999999999987e202 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)} \]

   if -3.0499999999999999e212 < x < 3.49999999999999987e202

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

Alternative 11: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right) + z\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* (log y) x) a) a (* y i))))
   (if (<= x -3.05e+212)
     t_1
     (if (<= x 3.5e+202)
       (+ (fma i y (+ (fma (log c) (- b 0.5) t) z)) a)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(((log(y) * x) / a), a, (y * i));
	double tmp;
	if (x <= -3.05e+212) {
		tmp = t_1;
	} else if (x <= 3.5e+202) {
		tmp = fma(i, y, (fma(log(c), (b - 0.5), t) + z)) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i))
	tmp = 0.0
	if (x <= -3.05e+212)
		tmp = t_1;
	elseif (x <= 3.5e+202)
		tmp = Float64(fma(i, y, Float64(fma(log(c), Float64(b - 0.5), t) + z)) + a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.05e+212], t$95$1, If[LessEqual[x, 3.5e+202], N[(N[(i * y + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0499999999999999e212 or 3.49999999999999987e202 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)} \]

   if -3.0499999999999999e212 < x < 3.49999999999999987e202

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

Alternative 12: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+202}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* (log y) x) a) a (* y i))))
   (if (<= x -3.05e+212)
     t_1
     (if (<= x 3.5e+202) (+ a (+ z (fma i y (* (log c) (- b 0.5))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(((log(y) * x) / a), a, (y * i));
	double tmp;
	if (x <= -3.05e+212) {
		tmp = t_1;
	} else if (x <= 3.5e+202) {
		tmp = a + (z + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i))
	tmp = 0.0
	if (x <= -3.05e+212)
		tmp = t_1;
	elseif (x <= 3.5e+202)
		tmp = Float64(a + Float64(z + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.05e+212], t$95$1, If[LessEqual[x, 3.5e+202], N[(a + N[(z + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0499999999999999e212 or 3.49999999999999987e202 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)} \]

   if -3.0499999999999999e212 < x < 3.49999999999999987e202

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 69.8

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

   7. Applied rewrites 69.8%

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

Alternative 13: 72.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, -0.5, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.36e-11)
   (fma (- b 0.5) (log c) (+ a (+ t z)))
   (+ (fma (log c) -0.5 (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.36e-11) {
		tmp = fma((b - 0.5), log(c), (a + (t + z)));
	} else {
		tmp = fma(log(c), -0.5, fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.36e-11)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)));
	else
		tmp = Float64(fma(log(c), -0.5, fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.36e-11], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * -0.5 + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e-11

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b - \frac{1}{2}, \log c, a + \left(t + \color{blue}{z}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 61.3

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

   9. Applied rewrites 61.3%

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

   if 1.36e-11 < y

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 84.3

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.3

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

   8. Applied rewrites 84.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.8%

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

Alternative 14: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot x}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{a}, a, y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+307)
     (fma y i (/ (* (* a (log y)) x) a))
     (if (<= t_1 4e+306)
       (fma (- b 0.5) (log c) (+ a (+ t z)))
       (fma (/ (* (log y) x) a) a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = fma(y, i, (((a * log(y)) * x) / a));
	} else if (t_1 <= 4e+306) {
		tmp = fma((b - 0.5), log(c), (a + (t + z)));
	} else {
		tmp = fma(((log(y) * x) / a), a, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = fma(y, i, Float64(Float64(Float64(a * log(y)) * x) / a));
	elseif (t_1 <= 4e+306)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)));
	else
		tmp = fma(Float64(Float64(log(y) * x) / a), a, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+307], N[(y * i + N[(N[(N[(a * N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+306], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] * a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5e307

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(\log y, x, t\right)\right) + z}{a}\right) \cdot a} + y \cdot i \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(x \cdot \log y\right)}}{a} + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{a \cdot \left(x \cdot \color{blue}{\log y}\right)}{a} + y \cdot i \]

      3. lower-log.f64 34.5

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

   8. Applied rewrites 34.5%

      \[\leadsto \frac{\color{blue}{a \cdot \left(x \cdot \log y\right)}}{a} + y \cdot i \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(x \cdot \log y\right)}{a} + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + \frac{a \cdot \left(x \cdot \log y\right)}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + \frac{a \cdot \left(x \cdot \log y\right)}{a} \]

      4. lower-fma.f64 34.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{a \cdot \left(x \cdot \log y\right)}{a}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \color{blue}{\left(x \cdot \log y\right)}}{a}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \left(x \cdot \color{blue}{\log y}\right)}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \left(\log y \cdot \color{blue}{x}\right)}{a}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot \color{blue}{x}}{a}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot \color{blue}{x}}{a}\right) \]

      10. lower-*.f64 34.4

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

   10. Applied rewrites 34.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot x}{a}\right)} \]

   if -5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.00000000000000007e306

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b - \frac{1}{2}, \log c, a + \left(t + \color{blue}{z}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 61.3

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

   9. Applied rewrites 61.3%

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

   if 4.00000000000000007e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{\color{blue}{a}} \cdot a + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

      3. lower-log.f64 32.0

         \[\leadsto \frac{x \cdot \log y}{a} \cdot a + y \cdot i \]

   6. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{x \cdot \log y}{a}} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a + y \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \log y}{a} \cdot a} + y \cdot i \]

      3. lower-fma.f64 32.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \log y}{a}, a, y \cdot i\right)} \]

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

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

Alternative 15: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot x}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+307)
     (fma y i (/ (* (* a (log y)) x) a))
     (if (<= t_1 1e+293)
       (fma (- b 0.5) (log c) (+ a (+ t z)))
       (fma y i (* 1.0 a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = fma(y, i, (((a * log(y)) * x) / a));
	} else if (t_1 <= 1e+293) {
		tmp = fma((b - 0.5), log(c), (a + (t + z)));
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = fma(y, i, Float64(Float64(Float64(a * log(y)) * x) / a));
	elseif (t_1 <= 1e+293)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+307], N[(y * i + N[(N[(N[(a * N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+293], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5e307

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(\log y, x, t\right)\right) + z}{a}\right) \cdot a} + y \cdot i \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(x \cdot \log y\right)}}{a} + y \cdot i \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{a \cdot \left(x \cdot \color{blue}{\log y}\right)}{a} + y \cdot i \]

      3. lower-log.f64 34.5

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

   8. Applied rewrites 34.5%

      \[\leadsto \frac{\color{blue}{a \cdot \left(x \cdot \log y\right)}}{a} + y \cdot i \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(x \cdot \log y\right)}{a} + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + \frac{a \cdot \left(x \cdot \log y\right)}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + \frac{a \cdot \left(x \cdot \log y\right)}{a} \]

      4. lower-fma.f64 34.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{a \cdot \left(x \cdot \log y\right)}{a}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \color{blue}{\left(x \cdot \log y\right)}}{a}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \left(x \cdot \color{blue}{\log y}\right)}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{a \cdot \left(\log y \cdot \color{blue}{x}\right)}{a}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot \color{blue}{x}}{a}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot \color{blue}{x}}{a}\right) \]

      10. lower-*.f64 34.4

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

   10. Applied rewrites 34.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{\left(a \cdot \log y\right) \cdot x}{a}\right)} \]

   if -5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.9999999999999992e292

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b - \frac{1}{2}, \log c, a + \left(t + \color{blue}{z}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 61.3

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

   9. Applied rewrites 61.3%

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

   if 9.9999999999999992e292 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 22500000000000:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 22500000000000.0)
   (fma (- b 0.5) (log c) (+ a (+ t z)))
   (fma y i (* 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 22500000000000.0) {
		tmp = fma((b - 0.5), log(c), (a + (t + z)));
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 22500000000000.0)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 22500000000000.0], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e13

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b - \frac{1}{2}, \log c, a + \left(t + \color{blue}{z}\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 61.3

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

   9. Applied rewrites 61.3%

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

   if 2.25e13 < y

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -40:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{a \cdot z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -40.0)
   (fma y i (/ (* a z) a))
   (fma y i (* 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -40.0) {
		tmp = fma(y, i, ((a * z) / a));
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -40.0)
		tmp = fma(y, i, Float64(Float64(a * z) / a));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], -40.0], N[(y * i + N[(N[(a * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -40

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(\log y, x, t\right)\right) + z}{a}\right) \cdot a} + y \cdot i \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{a \cdot z}}{a} + y \cdot i \]
   7. Step-by-step derivation

      1. lower-*.f64 33.9

         \[\leadsto \frac{a \cdot \color{blue}{z}}{a} + y \cdot i \]

   8. Applied rewrites 33.9%

      \[\leadsto \frac{\color{blue}{a \cdot z}}{a} + y \cdot i \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{a} + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + \frac{a \cdot z}{a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + \frac{a \cdot z}{a} \]

      4. lower-fma.f64 33.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{a \cdot z}{a}\right)} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \frac{a \cdot z}{a}\right)} \]

   if -40 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -40:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{z}{a} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -40.0)
   (fma y i (* (/ z a) a))
   (fma y i (* 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -40.0) {
		tmp = fma(y, i, ((z / a) * a));
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -40.0)
		tmp = fma(y, i, Float64(Float64(z / a) * a));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], -40.0], N[(y * i + N[(N[(z / a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -40

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\frac{z}{a}} \cdot a\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 31.5

          \[\leadsto \mathsf{fma}\left(y, i, \frac{z}{\color{blue}{a}} \cdot a\right) \]

   11. Applied rewrites 31.5%

       \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\frac{z}{a}} \cdot a\right) \]

   if -40 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 38.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ \mathbf{if}\;b \leq -9 \cdot 10^{+232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))))
   (if (<= b -9e+232) t_1 (if (<= b 4.8e+172) (fma y i (* 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double tmp;
	if (b <= -9e+232) {
		tmp = t_1;
	} else if (b <= 4.8e+172) {
		tmp = fma(y, i, (1.0 * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	tmp = 0.0
	if (b <= -9e+232)
		tmp = t_1;
	elseif (b <= 4.8e+172)
		tmp = fma(y, i, Float64(1.0 * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9e+232], t$95$1, If[LessEqual[b, 4.8e+172], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.9999999999999995e232 or 4.8000000000000001e172 < b

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \log c} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\log c} \]

      2. lower-log.f64 16.9

         \[\leadsto b \cdot \log c \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{b \cdot \log c} \]

   if -8.9999999999999995e232 < b < 4.8000000000000001e172

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -1.8e+153) (* x (log y)) (fma y i (* 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -1.8e+153) {
		tmp = x * log(y);
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -1.8e+153)
		tmp = Float64(x * log(y));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -1.8e+153], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e153

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.6

         \[\leadsto x \cdot \log y \]

   6. Applied rewrites 16.6%

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

   if -1.8e153 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lower-special-*.f64 N/A

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   5. Step-by-step derivation

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

      4. lower-fma.f64 38.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, 1 \cdot a\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (fma y i (* 1.0 a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(y, i, (1.0 * a));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(y, i, Float64(1.0 * a))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. 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. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. sum-to-mult N/A

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

   6. lower-special-*.f64 N/A

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

3. Applied rewrites 73.0%

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

4. Taylor expanded in a around inf

   \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
5. Step-by-step derivation

6. Applied rewrites 38.8%

   \[\leadsto \color{blue}{1} \cdot a + y \cdot i \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{1 \cdot a + y \cdot i} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot i + 1 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot i} + 1 \cdot a \]

   4. lower-fma.f64 38.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]

8. Applied rewrites 38.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, 1 \cdot a\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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)))