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

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

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

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

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, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (fma
  (- b 0.5)
  (log c)
  (+ (fma i y (+ (fmax z a) (fma (log y) x (fmin z a)))) 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, (fmax(z, a) + fma(log(y), x, fmin(z, a)))) + 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(fmax(z, a) + fma(log(y), x, fmin(z, a)))) + 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[(N[Max[z, a], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[Min[z, a], $MachinePrecision]), $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, 0.9× speedup

Math

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

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 (fmin z t))) a) (fmax z 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, fmin(z, t))) + a) + fmax(z, 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, fmin(z, t))) + a) + fmax(z, 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 + N[Min[z, t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] + N[Max[z, t], $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. 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: 99.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+
  (fmax (fmax z t) (fmax (fmin z t) a))
  (+ (fmin (fmin z 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) {
	return fmax(fmax(z, t), fmax(fmin(z, t), a)) + (fmin(fmin(z, t), a) + 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(fmax(fmax(z, t), fmax(fmin(z, t), a)) + Float64(fmin(fmin(z, t), a) + 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[(N[Max[N[Max[z, t], $MachinePrecision], N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]], $MachinePrecision] + N[(N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision] + 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 4: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, a + \left(t + \left(z + x \cdot \log y\right)\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, a + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (- b 0.5) (log c) (+ a (+ t (+ z (* x (log y))))))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 6e+172) (fma (- b 0.5) (log c) (+ (fma i y (+ a z)) 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((b - 0.5), log(c), (a + (t + (z + (x * log(y))))));
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 6e+172) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, (a + z)) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + Float64(z + Float64(x * log(y))))))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 6e+172)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, Float64(a + z)) + 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[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 6e+172], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(a + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 5.9999999999999998e172 < 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

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

      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 77.1

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

   6. Applied rewrites 77.1%

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

   if -8.9999999999999999e169 < x < 5.9999999999999998e172

   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 84.0%

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

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, \mathsf{max}\left(z, a\right) + \left(t + x \cdot \log y\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, \mathsf{max}\left(z, a\right) + \mathsf{min}\left(z, a\right)\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (- b 0.5) (log c) (+ (fmax z a) (+ t (* x (log y)))))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 2.9e+241)
       (fma (- b 0.5) (log c) (+ (fma i y (+ (fmax z a) (fmin 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((b - 0.5), log(c), (fmax(z, a) + (t + (x * log(y)))));
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 2.9e+241) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, (fmax(z, a) + fmin(z, a))) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(b - 0.5), log(c), Float64(fmax(z, a) + Float64(t + Float64(x * log(y)))))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 2.9e+241)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, Float64(fmax(z, a) + fmin(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[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[Max[z, a], $MachinePrecision] + N[(t + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 2.9e+241], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(N[Max[z, a], $MachinePrecision] + N[Min[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 2.9000000000000002e241 < 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

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

      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 77.1

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 61.4

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

   9. Applied rewrites 61.4%

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

   if -8.9999999999999999e169 < x < 2.9000000000000002e241

   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 84.0%

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

Alternative 6: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := t + \left(z + \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, a + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ t (+ z (fma x (log y) (* (log c) (- b 0.5)))))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 1.5e+243)
       (fma (- b 0.5) (log c) (+ (fma i y (+ a z)) 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 = t + (z + fma(x, log(y), (log(c) * (b - 0.5))));
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 1.5e+243) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, (a + z)) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t + Float64(z + fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 1.5e+243)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, Float64(a + z)) + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 1.5e+243], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(a + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 1.49999999999999992e243 < 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. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 62.3

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

   7. Applied rewrites 62.3%

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

   if -8.9999999999999999e169 < x < 1.49999999999999992e243

   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 84.0%

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

Alternative 7: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, a + z\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 1.5e+243)
       (fma (- b 0.5) (log c) (+ (fma i y (+ a z)) 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 = x * log(y);
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 1.5e+243) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, (a + z)) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 1.5e+243)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, Float64(a + z)) + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 1.5e+243], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(a + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 1.49999999999999992e243 < 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

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

      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 84.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.0

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 84.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 84.0

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

   8. Applied rewrites 84.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 16.8

          \[\leadsto x \cdot \log y \]

   11. Applied rewrites 16.8%

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

   if -8.9999999999999999e169 < x < 1.49999999999999992e243

   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 84.0%

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

Alternative 8: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \left(a + \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 1.5e+243)
       (+ (fma i y z) (+ a (fma (- b 0.5) (log c) 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 = x * log(y);
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 1.5e+243) {
		tmp = fma(i, y, z) + (a + fma((b - 0.5), log(c), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 1.5e+243)
		tmp = Float64(fma(i, y, z) + Float64(a + fma(Float64(b - 0.5), log(c), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 1.5e+243], N[(N[(i * y + z), $MachinePrecision] + N[(a + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 1.49999999999999992e243 < 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

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

      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 84.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.0

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 84.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 84.0

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

   8. Applied rewrites 84.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 16.8

          \[\leadsto x \cdot \log y \]

   11. Applied rewrites 16.8%

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

   if -8.9999999999999999e169 < x < 1.49999999999999992e243

   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 84.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.0

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 84.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 84.0

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

   8. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+r+ N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 84.0

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.0

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

   10. Applied rewrites 84.0%

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

Alternative 9: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(t, a\right) \leq 2.15 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{min}\left(t, a\right) + \left(z + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(i, y, \mathsf{max}\left(t, a\right) + z\right)\right) + \mathsf{min}\left(t, a\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (fmax t a) 2.15e+125)
   (+ (fmin t a) (+ z (fma i y (* (log c) (- b 0.5)))))
   (+ (fma -0.5 (log c) (fma i y (+ (fmax t a) z))) (fmin t a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (fmax(t, a) <= 2.15e+125) {
		tmp = fmin(t, a) + (z + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = fma(-0.5, log(c), fma(i, y, (fmax(t, a) + z))) + fmin(t, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (fmax(t, a) <= 2.15e+125)
		tmp = Float64(fmin(t, a) + Float64(z + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = Float64(fma(-0.5, log(c), fma(i, y, Float64(fmax(t, a) + z))) + fmin(t, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[Max[t, a], $MachinePrecision], 2.15e+125], N[(N[Min[t, a], $MachinePrecision] + N[(z + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(i * y + N[(N[Max[t, a], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Min[t, a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.15000000000000018e125

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 69.2

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

   7. Applied rewrites 69.2%

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

   if 2.15000000000000018e125 < a

   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 84.0%

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

   7. Taylor expanded in b around 0

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

   9. Applied rewrites 68.7%

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

       1. lift-fma.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. lower-fma.f64 68.7

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

   11. Applied rewrites 68.7%

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

Alternative 10: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(i, y, a + z\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (- b 0.5) (log c) (+ a (+ t z))))
        (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -1e+71)
     t_1
     (if (<= t_2 5e+227) (+ (fma -0.5 (log c) (fma i y (+ a z))) 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((b - 0.5), log(c), (a + (t + z)));
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -1e+71) {
		tmp = t_1;
	} else if (t_2 <= 5e+227) {
		tmp = fma(-0.5, log(c), fma(i, y, (a + z))) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -1e+71)
		tmp = t_1;
	elseif (t_2 <= 5e+227)
		tmp = Float64(fma(-0.5, log(c), fma(i, y, Float64(a + z))) + 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[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+71], t$95$1, If[LessEqual[t$95$2, 5e+227], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(i * y + N[(a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1e71 or 4.9999999999999996e227 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 77.1

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

   6. Applied rewrites 77.1%

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

   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 61.5

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

   9. Applied rewrites 61.5%

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

   if -1e71 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 4.9999999999999996e227

   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 84.0%

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

   7. Taylor expanded in b around 0

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

   9. Applied rewrites 68.7%

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

       1. lift-fma.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate-+r+ N/A

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

       4. lower-+.f64 N/A

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

       5. lower-fma.f64 68.7

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

   11. Applied rewrites 68.7%

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

Alternative 11: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := 1 \cdot \left(y \cdot i\right)\\ t_2 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 1.0 (* y i)))
        (t_2
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+306) (fma (- b 0.5) (log c) (+ a (+ t z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 1.0 * (y * i);
	double t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = fma((b - 0.5), log(c), (a + (t + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(1.0 * Float64(y * i))
	t_2 = 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_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(1.0 * N[(y * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+306], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)) < -inf.0 or 2.00000000000000003e306 < (+.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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right) \cdot \left(i \cdot y\right)} \]

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

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right) \cdot \left(i \cdot y\right)} \]

      7. lower-unsound-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right)} \cdot \left(i \cdot y\right) \]

      8. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{1 \cdot t}{i \cdot y}}\right) \cdot \left(i \cdot y\right) \]

      9. *-commutative N/A

         \[\leadsto \left(1 + \frac{1 \cdot t}{\color{blue}{y \cdot i}}\right) \cdot \left(i \cdot y\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(1 + \frac{1 \cdot t}{\color{blue}{y \cdot i}}\right) \cdot \left(i \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto \left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \color{blue}{\left(y \cdot i\right)} \]

      12. lift-*.f64 31.7

          \[\leadsto \left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \color{blue}{\left(y \cdot i\right)} \]

   8. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 24.2%

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

   if -inf.0 < (+.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)) < 2.00000000000000003e306

   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 77.1

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

   6. Applied rewrites 77.1%

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

   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 61.5

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

   9. Applied rewrites 61.5%

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

Alternative 12: 49.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + t\_1\right) + t\_3\right) + t\_4\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{t\_3}, t\_3, y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_4}{t\_3}, t\_3, y \cdot i\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fmax (fmax z t) t_2)))
   (if (<=
        (+
         (+ (+ (+ (+ (* x (log y)) t_1) t_3) t_4) (* (- b 0.5) (log c)))
         (* y i))
        -1e+36)
     (fma (/ t_1 t_3) t_3 (* y i))
     (fma (/ t_4 t_3) t_3 (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fmax(fmax(z, t), t_2);
	double tmp;
	if (((((((x * log(y)) + t_1) + t_3) + t_4) + ((b - 0.5) * log(c))) + (y * i)) <= -1e+36) {
		tmp = fma((t_1 / t_3), t_3, (y * i));
	} else {
		tmp = fma((t_4 / t_3), t_3, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fmax(fmax(z, t), t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + t_3) + t_4) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1e+36)
		tmp = fma(Float64(t_1 / t_3), t_3, Float64(y * i));
	else
		tmp = fma(Float64(t_4 / t_3), t_3, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -1e+36], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $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)) < -1.00000000000000004e36

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 32.1

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{t}}, t, y \cdot i\right) \]

   11. Applied rewrites 32.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, t, y \cdot i\right) \]

   if -1.00000000000000004e36 < (+.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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{t}}, t, y \cdot i\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 32.2

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

   11. Applied rewrites 32.2%

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

Alternative 13: 44.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), a\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{max}\left(\mathsf{max}\left(z, t\right), a\right)}{t\_3}, t\_3, y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c)))
        (t_2 (* (- b 0.5) (log c)))
        (t_3 (fmin (fmax z t) a)))
   (if (<= t_2 -1e+71)
     t_1
     (if (<= t_2 5e+227) (fma (/ (fmax (fmax z t) a) t_3) t_3 (* y i)) 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 t_2 = (b - 0.5) * log(c);
	double t_3 = fmin(fmax(z, t), a);
	double tmp;
	if (t_2 <= -1e+71) {
		tmp = t_1;
	} else if (t_2 <= 5e+227) {
		tmp = fma((fmax(fmax(z, t), a) / t_3), t_3, (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	t_3 = fmin(fmax(z, t), a)
	tmp = 0.0
	if (t_2 <= -1e+71)
		tmp = t_1;
	elseif (t_2 <= 5e+227)
		tmp = fma(Float64(fmax(fmax(z, t), a) / t_3), t_3, Float64(y * i));
	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]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], a], $MachinePrecision]}, If[LessEqual[t$95$2, -1e+71], t$95$1, If[LessEqual[t$95$2, 5e+227], N[(N[(N[Max[N[Max[z, t], $MachinePrecision], a], $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1e71 or 4.9999999999999996e227 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 16.6

         \[\leadsto b \cdot \log c \]

   4. Applied rewrites 16.6%

      \[\leadsto \color{blue}{b \cdot \log c} \]

   if -1e71 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 4.9999999999999996e227

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{t}}, t, y \cdot i\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 32.2

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

   11. Applied rewrites 32.2%

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

Alternative 14: 40.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(1, t, y \cdot i\right)\\ \mathbf{if}\;i \leq -1.15 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma 1.0 t (* y i))))
   (if (<= i -1.15e+20) t_1 (if (<= i 7.2e-46) (* b (log c)) 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(1.0, t, (y * i));
	double tmp;
	if (i <= -1.15e+20) {
		tmp = t_1;
	} else if (i <= 7.2e-46) {
		tmp = b * log(c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(1.0, t, Float64(y * i))
	tmp = 0.0
	if (i <= -1.15e+20)
		tmp = t_1;
	elseif (i <= 7.2e-46)
		tmp = Float64(b * log(c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(1.0 * t + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.15e+20], t$95$1, If[LessEqual[i, 7.2e-46], N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.15e20 or 7.2e-46 < 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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

   if -1.15e20 < i < 7.2e-46

   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 16.6

         \[\leadsto b \cdot \log c \]

   4. Applied rewrites 16.6%

      \[\leadsto \color{blue}{b \cdot \log c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -9 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{min}\left(t, a\right), y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -9e+169)
     t_1
     (if (<= x 1.5e+243) (fma 1.0 (fmin t a) (* y i)) t_1))))

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);
	double tmp;
	if (x <= -9e+169) {
		tmp = t_1;
	} else if (x <= 1.5e+243) {
		tmp = fma(1.0, fmin(t, a), (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -9e+169)
		tmp = t_1;
	elseif (x <= 1.5e+243)
		tmp = fma(1.0, fmin(t, a), Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+169], t$95$1, If[LessEqual[x, 1.5e+243], N[(1.0 * N[Min[t, a], $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e169 or 1.49999999999999992e243 < 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

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

      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 84.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 84.0

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 84.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 84.0

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

   8. Applied rewrites 84.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 16.8

          \[\leadsto x \cdot \log y \]

   11. Applied rewrites 16.8%

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

   if -8.9999999999999999e169 < x < 1.49999999999999992e243

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 73.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

Alternative 16: 37.0% accurate, 3.1× speedup

Math

\[\mathsf{fma}\left(1, \mathsf{min}\left(t, a\right), y \cdot i\right) \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (fma 1.0 (fmin t a) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(1.0, fmin(t, a), (y * i));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(1.0, fmin(t, a), Float64(y * i))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(1.0 * N[Min[t, a], $MachinePrecision] + N[(y * i), $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. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. sum-to-mult N/A

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

   8. lower-unsound-*.f64 N/A

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

3. Applied rewrites 73.8%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 37.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 37.7

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

8. Applied rewrites 37.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, t, y \cdot i\right)} \]
9. Add Preprocessing

Alternative 17: 24.2% accurate, 5.4× speedup

Math

\[1 \cdot \left(y \cdot i\right) \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* 1.0 (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 1.0 * (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 = 1.0d0 * (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 1.0 * (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return 1.0 * (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(1.0 * Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 1.0 * (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(1.0 * N[(y * i), $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. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. sum-to-mult N/A

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

   8. lower-unsound-*.f64 N/A

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

3. Applied rewrites 73.8%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 37.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right) \cdot \left(i \cdot y\right)} \]

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

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right) \cdot \left(i \cdot y\right)} \]

   7. lower-unsound-+.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{i \cdot y}\right)} \cdot \left(i \cdot y\right) \]

   8. lower-unsound-/.f64 N/A

      \[\leadsto \left(1 + \color{blue}{\frac{1 \cdot t}{i \cdot y}}\right) \cdot \left(i \cdot y\right) \]

   9. *-commutative N/A

      \[\leadsto \left(1 + \frac{1 \cdot t}{\color{blue}{y \cdot i}}\right) \cdot \left(i \cdot y\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(1 + \frac{1 \cdot t}{\color{blue}{y \cdot i}}\right) \cdot \left(i \cdot y\right) \]

   11. *-commutative N/A

       \[\leadsto \left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \color{blue}{\left(y \cdot i\right)} \]

   12. lift-*.f64 31.7

       \[\leadsto \left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \color{blue}{\left(y \cdot i\right)} \]

8. Applied rewrites 31.7%

   \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

9. Taylor expanded in y around inf

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

11. Applied rewrites 24.2%

    \[\leadsto \color{blue}{1} \cdot \left(y \cdot i\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(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)))