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

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

Alternatives: 20

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

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

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(a + Float64(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[(a + N[(t + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.4

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 99.4

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.4

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.4

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

   16. lift-+.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 45.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ t_2 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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)))
        (t_2 (fma y i (* (log y) x))))
   (if (<= t_1 -1e+306)
     t_2
     (if (<= t_1 -100.0)
       (fma (- b 0.5) (log c) z)
       (if (<= t_1 2e+299) (fma (- b 0.5) (log c) a) t_2)))))

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 t_2 = fma(y, i, (log(y) * x));
	double tmp;
	if (t_1 <= -1e+306) {
		tmp = t_2;
	} else if (t_1 <= -100.0) {
		tmp = fma((b - 0.5), log(c), z);
	} else if (t_1 <= 2e+299) {
		tmp = fma((b - 0.5), log(c), a);
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (t_1 <= -1e+306)
		tmp = t_2;
	elseif (t_1 <= -100.0)
		tmp = fma(Float64(b - 0.5), log(c), z);
	elseif (t_1 <= 2e+299)
		tmp = fma(Float64(b - 0.5), log(c), a);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+306], t$95$2, If[LessEqual[t$95$1, -100.0], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision], t$95$2]]]]]

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)) < -1.00000000000000002e306 or 2.0000000000000001e299 < (+.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 97.3%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 97.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.3

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 97.3

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

   4. Applied rewrites 97.3%

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

   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

   7. Applied rewrites 94.7%

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

   if -1.00000000000000002e306 < (+.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 99.8%

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

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 25.6

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

   4. Applied rewrites 25.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 13.9%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4

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

   9. Applied rewrites 41.4%

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

   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)) < 2.0000000000000001e299

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 26.3

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

   4. Applied rewrites 26.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 14.0%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 42.1

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

   9. Applied rewrites 42.1%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 34.0%

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

Alternative 3: 44.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+294)
     (fma y i z)
     (if (<= t_1 -100.0)
       (fma (- b 0.5) (log c) z)
       (if (<= t_1 2e+299) (fma (- b 0.5) (log c) a) (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 <= -5e+294) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -100.0) {
		tmp = fma((b - 0.5), log(c), z);
	} else if (t_1 <= 2e+299) {
		tmp = fma((b - 0.5), log(c), a);
	} 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 <= -5e+294)
		tmp = fma(y, i, z);
	elseif (t_1 <= -100.0)
		tmp = fma(Float64(b - 0.5), log(c), z);
	elseif (t_1 <= 2e+299)
		tmp = fma(Float64(b - 0.5), log(c), a);
	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, -5e+294], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $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)) < -4.9999999999999999e294

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.7%

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

   if -4.9999999999999999e294 < (+.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 99.8%

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

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 24.3

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

   4. Applied rewrites 24.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 12.1%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 42.5

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

   9. Applied rewrites 42.5%

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

   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)) < 2.0000000000000001e299

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 26.3

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

   4. Applied rewrites 26.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 14.0%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 42.1

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

   9. Applied rewrites 42.1%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 34.0%

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

   if 2.0000000000000001e299 < (+.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 95.8%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 95.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 83.6%

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

Alternative 4: 44.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(b, \log c, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+294)
     (fma y i z)
     (if (<= t_1 -2000000.0)
       (fma b (log c) z)
       (if (<= t_1 2e+299) (fma (- b 0.5) (log c) a) (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 <= -5e+294) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -2000000.0) {
		tmp = fma(b, log(c), z);
	} else if (t_1 <= 2e+299) {
		tmp = fma((b - 0.5), log(c), a);
	} 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 <= -5e+294)
		tmp = fma(y, i, z);
	elseif (t_1 <= -2000000.0)
		tmp = fma(b, log(c), z);
	elseif (t_1 <= 2e+299)
		tmp = fma(Float64(b - 0.5), log(c), a);
	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, -5e+294], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], N[(b * N[Log[c], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $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)) < -4.9999999999999999e294

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.7%

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

   if -4.9999999999999999e294 < (+.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)) < -2e6

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 25.1

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 12.5%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.2

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

   9. Applied rewrites 41.2%

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

   10. Taylor expanded in b around inf

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

   12. Applied rewrites 40.6%

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

   if -2e6 < (+.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)) < 2.0000000000000001e299

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 25.6

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

   4. Applied rewrites 25.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 13.6%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 43.2

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

   9. Applied rewrites 43.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 35.7%

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

   if 2.0000000000000001e299 < (+.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 95.8%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 95.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 83.6%

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

Alternative 5: 41.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(b, \log c, z\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+294)
     (fma y i z)
     (if (<= t_1 -2e+29)
       (fma b (log c) z)
       (if (<= t_1 200.0) (fma -0.5 (log c) 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 <= -5e+294) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -2e+29) {
		tmp = fma(b, log(c), z);
	} else if (t_1 <= 200.0) {
		tmp = fma(-0.5, log(c), 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 <= -5e+294)
		tmp = fma(y, i, z);
	elseif (t_1 <= -2e+29)
		tmp = fma(b, log(c), z);
	elseif (t_1 <= 200.0)
		tmp = fma(-0.5, log(c), 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, -5e+294], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -2e+29], N[(b * N[Log[c], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(-0.5 * N[Log[c], $MachinePrecision] + z), $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)) < -4.9999999999999999e294

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.7%

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

   if -4.9999999999999999e294 < (+.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)) < -1.99999999999999983e29

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 25.7

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

   4. Applied rewrites 25.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 12.8%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.0

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

   9. Applied rewrites 41.0%

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

   10. Taylor expanded in b around inf

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

   12. Applied rewrites 40.9%

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

   if -1.99999999999999983e29 < (+.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)) < 200

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 0.0%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 81.3

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 81.3%

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

   if 200 < (+.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 99.1%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 33.2%

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

Alternative 6: 28.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

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 1e+308) 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 <= 1e+308) {
		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 <= 1e+308) {
		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 <= 1e+308:
		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 <= 1e+308)
		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 <= 1e+308)
		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, 1e+308], 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 1e308 < (+.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 97.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.0%

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

   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 99.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 15.4%

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

   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)) < 1e308

   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 14.6%

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

Alternative 7: 57.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b - 0.5, t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 5e+63)
     (+ (fma i y z) (fma (log c) (- b 0.5) t))
     (if (<= t_1 2e+299) (fma (- b 0.5) (log c) a) (fma y i (* (log y) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= 5e+63) {
		tmp = fma(i, y, z) + fma(log(c), (b - 0.5), t);
	} else if (t_1 <= 2e+299) {
		tmp = fma((b - 0.5), log(c), a);
	} else {
		tmp = fma(y, i, (log(y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= 5e+63)
		tmp = Float64(fma(i, y, z) + fma(log(c), Float64(b - 0.5), t));
	elseif (t_1 <= 2e+299)
		tmp = fma(Float64(b - 0.5), log(c), a);
	else
		tmp = fma(y, i, Float64(log(y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+63], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5.00000000000000011e63

   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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.5%

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

   if 5.00000000000000011e63 < (+.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)) < 2.0000000000000001e299

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 26.2

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

   4. Applied rewrites 26.2%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 14.0%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.0

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

   9. Applied rewrites 41.0%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 32.3%

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

   if 2.0000000000000001e299 < (+.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 95.8%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 95.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   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

   7. Applied rewrites 91.9%

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

Alternative 8: 49.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

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 -2000000.0)
     (+ (fma i y z) (* (log c) b))
     (if (<= t_1 2e+299) (fma (- b 0.5) (log c) a) (fma y i (* (log y) x))))))

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 <= -2000000.0) {
		tmp = fma(i, y, z) + (log(c) * b);
	} else if (t_1 <= 2e+299) {
		tmp = fma((b - 0.5), log(c), a);
	} else {
		tmp = fma(y, i, (log(y) * x));
	}
	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 <= -2000000.0)
		tmp = Float64(fma(i, y, z) + Float64(log(c) * b));
	elseif (t_1 <= 2e+299)
		tmp = fma(Float64(b - 0.5), log(c), a);
	else
		tmp = fma(y, i, Float64(log(y) * x));
	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, -2000000.0], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2e6

   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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 57.6%

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

   if -2e6 < (+.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)) < 2.0000000000000001e299

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 25.6

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

   4. Applied rewrites 25.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 13.6%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 43.2

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

   9. Applied rewrites 43.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 35.7%

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

   if 2.0000000000000001e299 < (+.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 95.8%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 95.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   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

   7. Applied rewrites 91.9%

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

Alternative 9: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+29)
     (fma y i z)
     (if (<= t_1 200.0) (fma -0.5 (log c) 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+29) {
		tmp = fma(y, i, z);
	} else if (t_1 <= 200.0) {
		tmp = fma(-0.5, log(c), 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+29)
		tmp = fma(y, i, z);
	elseif (t_1 <= 200.0)
		tmp = fma(-0.5, log(c), 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+29], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(-0.5 * N[Log[c], $MachinePrecision] + z), $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)) < -1.99999999999999983e29

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 36.7%

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

   if -1.99999999999999983e29 < (+.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)) < 200

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 0.0%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 81.3

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 81.3%

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

   if 200 < (+.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 99.1%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 33.2%

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

Alternative 10: 34.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -1e+114) 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 <= -1e+114) {
		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 <= -1e+114)
		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, -1e+114], 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

   5. Applied rewrites 100.0%

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

   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)) < -1e114

   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 z around inf

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

   5. Applied rewrites 16.5%

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

   if -1e114 < (+.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 99.2%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.2

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 33.7%

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

Alternative 11: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i \leq -100:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b - 0.5, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\_1\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= (+ (+ (+ (+ (+ (* x (log y)) z) t) a) t_1) (* y i)) -100.0)
     (+ (fma i y z) (fma (log c) (- b 0.5) t))
     (+ (+ a t_1) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= -100.0) {
		tmp = fma(i, y, z) + fma(log(c), (b - 0.5), t);
	} else {
		tmp = (a + t_1) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + t_1) + Float64(y * i)) <= -100.0)
		tmp = Float64(fma(i, y, z) + fma(log(c), Float64(b - 0.5), t));
	else
		tmp = Float64(Float64(a + t_1) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(a + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.9%

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

   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 99.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 48.2%

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

Alternative 12: 37.6% accurate, 1.0× speedup

Math

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

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))
      -50.0)
   (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)) <= -50.0) {
		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)) <= -50.0)
		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], -50.0], 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)) < -50

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 35.4%

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

   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 99.1%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 32.8%

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

Alternative 13: 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 99.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 14.2%

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

   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 99.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 12.6%

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

Alternative 14: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(i, y, z\right)\right)\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+213}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, i \cdot y\right)\right) + z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.55e+94)
   (fma (log c) (- b 0.5) (fma (log y) x (fma i y z)))
   (if (<= a 7e+213)
     (+ (+ (+ (fma -0.5 (log c) (fma (log y) x (* i y))) z) t) a)
     (fma y i (fma (log c) (- b 0.5) (+ a (+ t z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.55e+94) {
		tmp = fma(log(c), (b - 0.5), fma(log(y), x, fma(i, y, z)));
	} else if (a <= 7e+213) {
		tmp = ((fma(-0.5, log(c), fma(log(y), x, (i * y))) + z) + t) + a;
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + (t + z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.55e+94)
		tmp = fma(log(c), Float64(b - 0.5), fma(log(y), x, fma(i, y, z)));
	elseif (a <= 7e+213)
		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), Float64(b - 0.5), Float64(a + Float64(t + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.55e+94], N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7e+213], 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] * N[(b - 0.5), $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.54999999999999996e94

   1. Initial program 99.4%

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.3%

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

   if 1.54999999999999996e94 < a < 6.9999999999999994e213

   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 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

   5. Applied rewrites 82.3%

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

   if 6.9999999999999994e213 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.6%

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

Alternative 15: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(i, y, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.55e+149)
   (fma (log c) (- b 0.5) (fma (log y) x (fma i y z)))
   (fma y i (fma (log c) (- b 0.5) (+ a (+ t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.55e+149) {
		tmp = fma(log(c), (b - 0.5), fma(log(y), x, fma(i, y, z)));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + (t + z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.55e+149)
		tmp = fma(log(c), Float64(b - 0.5), fma(log(y), x, fma(i, y, z)));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + Float64(t + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.55e+149], N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.54999999999999993e149

   1. Initial program 99.4%

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.6%

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

   if 1.54999999999999993e149 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 91.3%

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

Alternative 16: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + t\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -1.75e+216)
   (+ (fma (log y) x (fma (log c) (- b 0.5) z)) t)
   (if (<= x 6.6e+136)
     (fma y i (fma (log c) (- b 0.5) (+ a (+ t z))))
     (+ (fma i y z) (* (log y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -1.75e+216) {
		tmp = fma(log(y), x, fma(log(c), (b - 0.5), z)) + t;
	} else if (x <= 6.6e+136) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + (t + z))));
	} else {
		tmp = fma(i, y, z) + (log(y) * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -1.75e+216)
		tmp = Float64(fma(log(y), x, fma(log(c), Float64(b - 0.5), z)) + t);
	elseif (x <= 6.6e+136)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + Float64(t + z))));
	else
		tmp = Float64(fma(i, y, z) + Float64(log(y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -1.75e+216], N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[x, 6.6e+136], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.74999999999999996e216

   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 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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.0%

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

   if -1.74999999999999996e216 < x < 6.59999999999999984e136

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.1%

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

   if 6.59999999999999984e136 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in 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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.2%

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

Alternative 17: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, t\_1\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -2.5e+205)
     (fma (- b 0.5) (log c) t_1)
     (if (<= x 6.6e+136)
       (fma y i (fma (log c) (- b 0.5) (+ a (+ t z))))
       (+ (fma i y 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 = log(y) * x;
	double tmp;
	if (x <= -2.5e+205) {
		tmp = fma((b - 0.5), log(c), t_1);
	} else if (x <= 6.6e+136) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + (t + z))));
	} else {
		tmp = fma(i, y, z) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -2.5e+205)
		tmp = fma(Float64(b - 0.5), log(c), t_1);
	elseif (x <= 6.6e+136)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + Float64(t + z))));
	else
		tmp = Float64(fma(i, y, z) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.5e+205], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[x, 6.6e+136], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e205

   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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. add-sqr-sqrt N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-+.f64 27.9

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

   4. Applied rewrites 27.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 4.7%

      \[\leadsto \mathsf{fma}\left(\log c \cdot \sqrt{b - 0.5}, \sqrt{b - 0.5}, \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 12.7

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

   9. Applied rewrites 12.7%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 80.3%

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

   if -2.5000000000000001e205 < x < 6.59999999999999984e136

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.6%

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

   if 6.59999999999999984e136 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in 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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.2%

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

Alternative 18: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+216} \lor \neg \left(x \leq 6.6 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.1e+216) (not (<= x 6.6e+136)))
   (+ (fma i y z) (* (log y) x))
   (fma y i (fma (log c) (- b 0.5) (+ a (+ t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.1e+216) || !(x <= 6.6e+136)) {
		tmp = fma(i, y, z) + (log(y) * x);
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + (t + z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.1e+216) || !(x <= 6.6e+136))
		tmp = Float64(fma(i, y, z) + Float64(log(y) * x));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + Float64(t + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.1e+216], N[Not[LessEqual[x, 6.6e+136]], $MachinePrecision]], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e216 or 6.59999999999999984e136 < x

   1. Initial program 98.1%

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.5%

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

   if -1.1e216 < x < 6.59999999999999984e136

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+216} \lor \neg \left(x \leq 6.6 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + \left(t + z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+216} \lor \neg \left(x \leq 6.6 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.1e+216) (not (<= x 6.6e+136)))
   (+ (fma i y z) (* (log y) x))
   (+ (+ a t) (fma (log c) (- b 0.5) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.1e+216) || !(x <= 6.6e+136)) {
		tmp = fma(i, y, z) + (log(y) * x);
	} else {
		tmp = (a + t) + fma(log(c), (b - 0.5), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.1e+216) || !(x <= 6.6e+136))
		tmp = Float64(fma(i, y, z) + Float64(log(y) * x));
	else
		tmp = Float64(Float64(a + t) + fma(log(c), Float64(b - 0.5), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.1e+216], N[Not[LessEqual[x, 6.6e+136]], $MachinePrecision]], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e216 or 6.59999999999999984e136 < x

   1. Initial program 98.1%

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.5%

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

   if -1.1e216 < x < 6.59999999999999984e136

   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

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

   5. Applied rewrites 95.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+216} \lor \neg \left(x \leq 6.6 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 16.5% 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 99.4%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 14.9%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(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)))