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

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

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), ((a + t) + fma(log(y), x, z))));
}

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 56.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+300)
     (fma y i (* (log y) x))
     (if (<= t_1 -2e+21)
       (+ (+ t z) (* (log c) b))
       (if (<= t_1 500.0)
         (fma y i (fma (log c) -0.5 z))
         (fma y i (fma (log c) b a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = fma(y, i, (log(y) * x));
	} else if (t_1 <= -2e+21) {
		tmp = (t + z) + (log(c) * b);
	} else if (t_1 <= 500.0) {
		tmp = fma(y, i, fma(log(c), -0.5, z));
	} else {
		tmp = fma(y, i, fma(log(c), b, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = fma(y, i, Float64(log(y) * x));
	elseif (t_1 <= -2e+21)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	elseif (t_1 <= 500.0)
		tmp = fma(y, i, fma(log(c), -0.5, z));
	else
		tmp = fma(y, i, fma(log(c), b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+21], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 500.0], N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)) < -2.0000000000000001e300

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -2.0000000000000001e300 < (+.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)) < -2e21

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 53.3

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 53.3%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -2e21 < (+.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)) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 86.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 86.7

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 83.6%

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

   if 500 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative 54.5

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

      2. *-commutative 54.5

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

      3. +-commutative 54.5

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

      4. associate-+l+ 54.5

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 54.3%

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

Alternative 3: 49.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+300)
     (fma y i (* (log y) x))
     (if (<= t_1 -2e+21)
       (+ (+ t z) (* (log c) b))
       (if (<= t_1 500.0) (fma y i (fma (log c) -0.5 z)) (fma y i a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = fma(y, i, (log(y) * x));
	} else if (t_1 <= -2e+21) {
		tmp = (t + z) + (log(c) * b);
	} else if (t_1 <= 500.0) {
		tmp = fma(y, i, fma(log(c), -0.5, z));
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = fma(y, i, Float64(log(y) * x));
	elseif (t_1 <= -2e+21)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	elseif (t_1 <= 500.0)
		tmp = fma(y, i, fma(log(c), -0.5, z));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+21], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 500.0], N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + z), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)) < -2.0000000000000001e300

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -2.0000000000000001e300 < (+.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)) < -2e21

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 53.3

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 53.3%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -2e21 < (+.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)) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 86.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 86.7

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 83.6%

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

   if 500 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.7

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

      2. +-commutative 39.7

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

      3. *-commutative 39.7

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

      4. +-commutative 39.7

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

      5. associate-+l+ 39.7

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

      6. +-commutative 39.7

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

   7. Applied rewrites 39.7%

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

Alternative 4: 28.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* i y)
     (if (<= t_1 -50.0) z (if (<= t_1 2e+300) a (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -50.0) {
		tmp = z;
	} else if (t_1 <= 2e+300) {
		tmp = a;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = i * y;
	} else if (t_1 <= -50.0) {
		tmp = z;
	} else if (t_1 <= 2e+300) {
		tmp = a;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = i * y
	elif t_1 <= -50.0:
		tmp = z
	elif t_1 <= 2e+300:
		tmp = a
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -50.0)
		tmp = z;
	elseif (t_1 <= 2e+300)
		tmp = a;
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = i * y;
	elseif (t_1 <= -50.0)
		tmp = z;
	elseif (t_1 <= 2e+300)
		tmp = a;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -50.0], z, If[LessEqual[t$95$1, 2e+300], a, N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0 or 2.0000000000000001e300 < (+.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 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 18.8%

      \[\leadsto \color{blue}{z} \]

   if -50 < (+.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)) < 2.0000000000000001e300

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 18.6%

      \[\leadsto \color{blue}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+300)
     (fma y i (* (log y) x))
     (if (<= t_1 -50.0) (+ (+ t z) (* (log c) b)) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = fma(y, i, (log(y) * x));
	} else if (t_1 <= -50.0) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = fma(y, i, Float64(log(y) * x));
	elseif (t_1 <= -50.0)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -50.0], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.0000000000000001e300

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -2.0000000000000001e300 < (+.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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 52.5

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.5%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -50 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 6: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ t_2 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, i, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -50:\\ \;\;\;\;\left(t + z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b))
        (t_2
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_2 (- INFINITY))
     (fma y i t_1)
     (if (<= t_2 -50.0) (+ (+ t z) t_1) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(y, i, t_1);
	} else if (t_2 <= -50.0) {
		tmp = (t + z) + t_1;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	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 = fma(y, i, t_1);
	elseif (t_2 <= -50.0)
		tmp = Float64(Float64(t + z) + t_1);
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $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)], N[(y * i + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -50.0], N[(N[(t + z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, b \cdot \log c\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, b \cdot \log c\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, b \cdot \log c\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, b \cdot \log c\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, i, b \cdot \log c\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{b} \cdot \log c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \log c \cdot \color{blue}{b}\right) \]

      8. lift-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, i, \log c \cdot b\right) \]

      9. lift-*.f64 97.0

         \[\leadsto \mathsf{fma}\left(y, i, \log c \cdot \color{blue}{b}\right) \]

   7. Applied rewrites 97.0%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot b}\right) \]

   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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 52.7

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.7%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -50 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 7: 47.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+294)
     (fma y i z)
     (if (<= t_1 -50.0) (+ (+ t z) (* (log c) b)) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+294) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -50.0) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+294)
		tmp = fma(y, i, z);
	elseif (t_1 <= -50.0)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+294], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -50.0], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.00000000000000027e294

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      2. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      3. *-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      4. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      5. associate-+l+ 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      6. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

   7. Applied rewrites 67.5%

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

   if -4.00000000000000027e294 < (+.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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 52.0

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.0%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -50 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 8: 40.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;z + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+294)
     (fma y i z)
     (if (<= t_1 -50.0) (+ z (* (log c) b)) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+294) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -50.0) {
		tmp = z + (log(c) * b);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+294)
		tmp = fma(y, i, z);
	elseif (t_1 <= -50.0)
		tmp = Float64(z + Float64(log(c) * b));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+294], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -50.0], N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.00000000000000027e294

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      2. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      3. *-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      4. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      5. associate-+l+ 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      6. +-commutative 67.5

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

   7. Applied rewrites 67.5%

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

   if -4.00000000000000027e294 < (+.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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 52.0

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.0%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   9. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log c} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 34.3%

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

   if -50 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 9: 33.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY)) (* i y) (if (<= t_1 -50.0) z (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -50.0) {
		tmp = z;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -50.0)
		tmp = z;
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -50.0], z, N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 18.8%

      \[\leadsto \color{blue}{z} \]

   if -50 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 10: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + t\right) + z\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, t\_1\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, i \cdot y\right)\right) + z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ a t) z)) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+15)
     (fma y i (fma (log c) (- b 0.5) t_1))
     (if (<= t_2 5e+146)
       (+ (+ (+ (fma -0.5 (log c) (fma (log y) x (* i y))) z) t) a)
       (fma y i (fma (log c) b t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + t) + z;
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+15) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), t_1));
	} else if (t_2 <= 5e+146) {
		tmp = ((fma(-0.5, log(c), fma(log(y), x, (i * y))) + z) + t) + a;
	} else {
		tmp = fma(y, i, fma(log(c), b, t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + t) + z)
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+15)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), t_1));
	elseif (t_2 <= 5e+146)
		tmp = Float64(Float64(Float64(fma(-0.5, log(c), fma(log(y), x, Float64(i * y))) + z) + t) + a);
	else
		tmp = fma(y, i, fma(log(c), b, t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+15], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+146], N[(N[(N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * b + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -2e15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.2%

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

   if -2e15 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 4.9999999999999999e146

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 98.0%

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

   if 4.9999999999999999e146 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 90.9%

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

Alternative 11: 54.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 300:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      300.0)
   (fma y i (fma (log c) (- b 0.5) z))
   (fma y i (fma (log c) (- b 0.5) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 300.0) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 300.0)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 300.0], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 300

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 54.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 54.1

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 54.2

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

   7. Applied rewrites 54.2%

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

   if 300 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative 54.5

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

      2. *-commutative 54.5

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

      3. +-commutative 54.5

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

      4. associate-+l+ 54.5

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

   7. Applied rewrites 54.5%

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

Alternative 12: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -100:\\ \;\;\;\;\left(t + z\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -100.0)
   (+ (+ t z) (* i y))
   (fma y i a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0) {
		tmp = (t + z) + (i * y);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -100.0)
		tmp = Float64(Float64(t + z) + Float64(i * y));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(t + z), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $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)) < -100

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 53.5

         \[\leadsto \left(t + z\right) + i \cdot y \]

   8. Applied rewrites 53.5%

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

   if -100 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 13: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -100:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -100.0)
   (fma y i z)
   (fma y i a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0) {
		tmp = fma(y, i, z);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -100.0)
		tmp = fma(y, i, z);
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -100.0], N[(y * i + z), $MachinePrecision], N[(y * i + a), $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)) < -100

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      2. +-commutative 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      3. *-commutative 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      4. +-commutative 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      5. associate-+l+ 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

      6. +-commutative 38.0

         \[\leadsto \mathsf{fma}\left(y, i, z\right) \]

   7. Applied rewrites 38.0%

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

   if -100 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 39.1

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

      2. +-commutative 39.1

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

      3. *-commutative 39.1

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

      4. +-commutative 39.1

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

      5. associate-+l+ 39.1

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

      6. +-commutative 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 14: 16.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   z
   a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-50.0d0)) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -50.0:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -50.0)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], z, a]

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)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{z} \]

   if -50 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + t\right) + \left(\mathsf{fma}\left(\log y, x, z\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ a t) (+ (fma (log y) x z) (* (log c) (- b 0.5))))))
   (if (<= x -1.3e+118)
     t_1
     (if (<= x 5.5e+145)
       (fma y i (fma (log c) (- b 0.5) (+ (+ 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 = (a + t) + (fma(log(y), x, z) + (log(c) * (b - 0.5)));
	double tmp;
	if (x <= -1.3e+118) {
		tmp = t_1;
	} else if (x <= 5.5e+145) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((a + t) + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + t) + Float64(fma(log(y), x, z) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (x <= -1.3e+118)
		tmp = t_1;
	elseif (x <= 5.5e+145)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(a + 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[(N[(a + t), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+118], t$95$1, If[LessEqual[x, 5.5e+145], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000008e118 or 5.4999999999999995e145 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -1.30000000000000008e118 < x < 5.4999999999999995e145

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.3%

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

Alternative 16: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z + \mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, i \cdot y\right)\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ z (fma -0.5 (log c) (fma (log y) x (* i y))))))
   (if (<= x -6.2e+155)
     t_1
     (if (<= x 1.4e+210)
       (fma y i (fma (log c) (- b 0.5) (+ (+ 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 = z + fma(-0.5, log(c), fma(log(y), x, (i * y)));
	double tmp;
	if (x <= -6.2e+155) {
		tmp = t_1;
	} else if (x <= 1.4e+210) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((a + t) + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(z + fma(-0.5, log(c), fma(log(y), x, Float64(i * y))))
	tmp = 0.0
	if (x <= -6.2e+155)
		tmp = t_1;
	elseif (x <= 1.4e+210)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(a + 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[(z + N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+155], t$95$1, If[LessEqual[x, 1.4e+210], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999978e155 or 1.4000000000000001e210 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lower-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 78.3%

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

   if -6.19999999999999978e155 < x < 1.4000000000000001e210

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.9%

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

Alternative 17: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+127}:\\ \;\;\;\;\left(t + z\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.08e+127)
   (+ (+ t z) (fma i y (fma (log c) (- b 0.5) (* (log y) x))))
   (+ (+ (+ (fma (log c) (- b 0.5) z) t) a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.08e+127) {
		tmp = (t + z) + fma(i, y, fma(log(c), (b - 0.5), (log(y) * x)));
	} else {
		tmp = ((fma(log(c), (b - 0.5), z) + t) + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.08e+127)
		tmp = Float64(Float64(t + z) + fma(i, y, fma(log(c), Float64(b - 0.5), Float64(log(y) * x))));
	else
		tmp = Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), z) + t) + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.08e+127], N[(N[(t + z), $MachinePrecision] + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.08000000000000001e127

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 1.08000000000000001e127 < 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 90.5

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

   5. Applied rewrites 90.5%

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

Alternative 18: 90.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + z\right) + \log y \cdot x\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ t z) (* (log y) x))))
   (if (<= x -1.2e+207)
     t_1
     (if (<= x 3.3e+215)
       (fma y i (fma (log c) (- b 0.5) (+ (+ 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 = (t + z) + (log(y) * x);
	double tmp;
	if (x <= -1.2e+207) {
		tmp = t_1;
	} else if (x <= 3.3e+215) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), ((a + t) + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + z) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -1.2e+207)
		tmp = t_1;
	elseif (x <= 3.3e+215)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(Float64(a + 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[(N[(t + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+207], t$95$1, If[LessEqual[x, 3.3e+215], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e207 or 3.2999999999999998e215 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

   if -1.2e207 < x < 3.2999999999999998e215

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 93.6%

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

Alternative 19: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + z\right) + \log y \cdot x\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, \left(a + t\right) + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ t z) (* (log y) x))))
   (if (<= x -1.2e+207)
     t_1
     (if (<= x 3.3e+215) (fma y i (fma (log c) b (+ (+ 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 = (t + z) + (log(y) * x);
	double tmp;
	if (x <= -1.2e+207) {
		tmp = t_1;
	} else if (x <= 3.3e+215) {
		tmp = fma(y, i, fma(log(c), b, ((a + t) + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + z) + Float64(log(y) * x))
	tmp = 0.0
	if (x <= -1.2e+207)
		tmp = t_1;
	elseif (x <= 3.3e+215)
		tmp = fma(y, i, fma(log(c), b, Float64(Float64(a + 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[(N[(t + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+207], t$95$1, If[LessEqual[x, 3.3e+215], N[(y * i + N[(N[Log[c], $MachinePrecision] * b + N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e207 or 3.2999999999999998e215 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

   if -1.2e207 < x < 3.2999999999999998e215

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 93.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 91.9%

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

Alternative 20: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\left(t + z\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -2.8e+157) (+ (+ t z) (* i y)) (fma y i (fma (log c) (- b 0.5) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -2.8e+157) {
		tmp = (t + z) + (i * y);
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -2.8e+157)
		tmp = Float64(Float64(t + z) + Float64(i * y));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -2.8e+157], N[(N[(t + z), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000003e157

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 92.6

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\left(t + z\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(t + z\right) + i \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. lower-*.f64 75.1

         \[\leadsto \left(t + z\right) + i \cdot y \]

   8. Applied rewrites 75.1%

      \[\leadsto \left(t + z\right) + i \cdot \color{blue}{y} \]

   if -2.8000000000000003e157 < z

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. lift-log.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      8. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right)} \cdot \log c\right) + y \cdot i \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c}\right) + y \cdot i \]

      10. lift-log.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \color{blue}{\log c}\right) + y \cdot i \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot i + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot i} + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + \mathsf{fma}\left(\log y, x, z\right)\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - \frac{1}{2}, \color{blue}{a}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 57.3

         \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right) \]

      2. *-commutative 57.3

         \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right) \]

      3. +-commutative 57.3

         \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right) \]

      4. associate-+l+ 57.3

         \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right) \]

   7. Applied rewrites 57.3%

      \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \color{blue}{a}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 16.3% accurate, 234.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 a)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return a

Julia

function code(x, y, z, t, a, b, c, i)
	return a
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

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. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
4. Step-by-step derivation

5. Applied rewrites 16.3%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(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)))