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

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

Alternatives: 23

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 90.5% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-log.f64 84.1

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

4. Applied rewrites 84.1%

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

Alternative 3: 90.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b - 0.5, z\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+126}:\\ \;\;\;\;\left(\frac{t\_1}{x} + \log y\right) \cdot x + a\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(t\_1 + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, \mathsf{fma}\left(x, \log y, z\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) (- b 0.5) z)))
   (if (<= x -2.65e+126)
     (+ (* (+ (/ t_1 x) (log y)) x) a)
     (if (<= x 4e+115)
       (fma y i (+ (+ t_1 t) a))
       (+ (fma (log c) b (fma x (log y) z)) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), (b - 0.5), z);
	double tmp;
	if (x <= -2.65e+126) {
		tmp = (((t_1 / x) + log(y)) * x) + a;
	} else if (x <= 4e+115) {
		tmp = fma(y, i, ((t_1 + t) + a));
	} else {
		tmp = fma(log(c), b, fma(x, log(y), z)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), Float64(b - 0.5), z)
	tmp = 0.0
	if (x <= -2.65e+126)
		tmp = Float64(Float64(Float64(Float64(t_1 / x) + log(y)) * x) + a);
	elseif (x <= 4e+115)
		tmp = fma(y, i, Float64(Float64(t_1 + t) + a));
	else
		tmp = Float64(fma(log(c), b, fma(x, log(y), z)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[x, -2.65e+126], N[(N[(N[(N[(t$95$1 / x), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 4e+115], N[(y * i + N[(N[(t$95$1 + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * b + N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65000000000000014e126

   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. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 82.9

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 73.0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   10. Applied rewrites 72.9%

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

   if -2.65000000000000014e126 < x < 4.0000000000000001e115

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 97.6

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

   6. Applied rewrites 97.6%

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

   if 4.0000000000000001e115 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 71.1%

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

Alternative 4: 89.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b, \mathsf{fma}\left(x, \log y, z\right)\right) + a\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma (log c) b (fma x (log y) z)) a)))
   (if (<= x -2.65e+126)
     t_1
     (if (<= x 4e+115) (fma y i (+ (+ (fma (log c) (- b 0.5) z) t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), b, fma(x, log(y), z)) + a;
	double tmp;
	if (x <= -2.65e+126) {
		tmp = t_1;
	} else if (x <= 4e+115) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + t) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(log(c), b, fma(x, log(y), z)) + a)
	tmp = 0.0
	if (x <= -2.65e+126)
		tmp = t_1;
	elseif (x <= 4e+115)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + t) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[Log[c], $MachinePrecision] * b + N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[x, -2.65e+126], t$95$1, If[LessEqual[x, 4e+115], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.65000000000000014e126 or 4.0000000000000001e115 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 72.0

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

   7. Applied rewrites 72.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 72.0%

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

   if -2.65000000000000014e126 < x < 4.0000000000000001e115

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 97.6

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

   6. Applied rewrites 97.6%

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

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.85 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, t + \mathsf{fma}\left(\log y, x, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.85e+115)
   (fma y i (fma (log c) (- b 0.5) (+ t (fma (log y) x z))))
   (fma y i (+ (+ (fma (log c) (- b 0.5) z) t) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.85e+115) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (t + fma(log(y), x, z))));
	} else {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + t) + a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.85e+115)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(t + fma(log(y), x, z))));
	else
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + t) + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.85e+115], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(t + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.84999999999999983e115

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 90.1%

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

   if 2.84999999999999983e115 < a

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 90.3

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

   6. Applied rewrites 90.3%

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

Alternative 6: 84.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (log y) x))))
   (if (<= x -4e+198)
     t_1
     (if (<= x 7.8e+195)
       (fma y i (+ (+ (fma (log c) (- b 0.5) z) t) a))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (log(y) * x));
	double tmp;
	if (x <= -4e+198) {
		tmp = t_1;
	} else if (x <= 7.8e+195) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + t) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (x <= -4e+198)
		tmp = t_1;
	elseif (x <= 7.8e+195)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + t) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+198], t$95$1, If[LessEqual[x, 7.8e+195], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000007e198 or 7.7999999999999995e195 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if -4.00000000000000007e198 < x < 7.7999999999999995e195

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 94.4

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

   6. Applied rewrites 94.4%

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

Alternative 7: 76.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(z + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (log y) x))))
   (if (<= x -4e+198)
     t_1
     (if (<= x 7.8e+195) (+ (+ (+ z a) (* (- b 0.5) (log c))) (* y i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (log(y) * x));
	double tmp;
	if (x <= -4e+198) {
		tmp = t_1;
	} else if (x <= 7.8e+195) {
		tmp = ((z + a) + ((b - 0.5) * log(c))) + (y * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (x <= -4e+198)
		tmp = t_1;
	elseif (x <= 7.8e+195)
		tmp = Float64(Float64(Float64(z + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+198], t$95$1, If[LessEqual[x, 7.8e+195], N[(N[(N[(z + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000007e198 or 7.7999999999999995e195 < x

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if -4.00000000000000007e198 < x < 7.7999999999999995e195

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 77.1%

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

Alternative 8: 64.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t + a\right) + t\_1\right) + y \cdot i\\ \end{array} \end{array} \]

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)) 2e+18)
     (fma y i (fma (log c) (- b 0.5) z))
     (+ (+ (+ 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)) <= 2e+18) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = ((t + 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)) <= 2e+18)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = Float64(Float64(Float64(t + 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], 2e+18], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t + a), $MachinePrecision] + 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)) < 2e18

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative 54.6

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

      2. *-commutative 54.6

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

      3. +-commutative 54.6

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

      4. associate-+l+ 54.6

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

   6. Applied rewrites 54.6%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 68.7%

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

Alternative 9: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      2e+18)
   (fma y i (fma (log c) (- b 0.5) z))
   (fma y i (fma (log c) (- b 0.5) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 2e+18) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 2e+18)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 2e+18], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2e18

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative 54.6

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

      2. *-commutative 54.6

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

      3. +-commutative 54.6

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

      4. associate-+l+ 54.6

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

   6. Applied rewrites 54.6%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in a around inf

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

      1. +-commutative 53.8

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

      2. *-commutative 53.8

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

      3. +-commutative 53.8

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

      4. associate-+l+ 53.8

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

   6. Applied rewrites 53.8%

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

Alternative 10: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot \log y + z\right) + t\right) + a\\ t_2 := \left(t\_1 + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_2 \leq -20000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (+ (* x (log y)) z) t) a))
        (t_2 (+ (+ t_1 (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 (- INFINITY))
     (* i y)
     (if (<= t_2 -20000.0) t_1 (fma y i (fma (log c) (- b 0.5) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((x * log(y)) + z) + t) + a;
	double t_2 = (t_1 + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_2 <= -20000.0) {
		tmp = t_1;
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a)
	t_2 = Float64(Float64(t_1 + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_2 <= -20000.0)
		tmp = t_1;
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$2, -20000.0], t$95$1, N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

   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)) < -2e4

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 86.8

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

      2. lift-log.f64 69.0

         \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

   7. Applied rewrites 69.0%

      \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

   if -2e4 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in a around inf

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

      1. +-commutative 55.0

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

      2. *-commutative 55.0

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

      3. +-commutative 55.0

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

      4. associate-+l+ 55.0

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

   6. Applied rewrites 55.0%

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

Alternative 11: 54.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot \log y + z\right) + t\right) + a\\ \mathbf{if}\;y \leq 1.32 \cdot 10^{-185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (+ (* x (log y)) z) t) a)))
   (if (<= y 1.32e-185)
     t_1
     (if (<= y 1.55e-109)
       (+ (fma (log c) (- b 0.5) z) a)
       (if (<= y 1.9e+150) t_1 (fma y i z))))))

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;
	double tmp;
	if (y <= 1.32e-185) {
		tmp = t_1;
	} else if (y <= 1.55e-109) {
		tmp = fma(log(c), (b - 0.5), z) + a;
	} else if (y <= 1.9e+150) {
		tmp = t_1;
	} else {
		tmp = fma(y, i, z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a)
	tmp = 0.0
	if (y <= 1.32e-185)
		tmp = t_1;
	elseif (y <= 1.55e-109)
		tmp = Float64(fma(log(c), Float64(b - 0.5), z) + a);
	elseif (y <= 1.9e+150)
		tmp = t_1;
	else
		tmp = fma(y, i, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[y, 1.32e-185], t$95$1, If[LessEqual[y, 1.55e-109], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[y, 1.9e+150], t$95$1, N[(y * i + z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.32e-185 or 1.55e-109 < y < 1.89999999999999995e150

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

      2. lift-log.f64 66.4

         \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

   7. Applied rewrites 66.4%

      \[\leadsto \left(\left(x \cdot \log y + z\right) + t\right) + a \]

   if 1.32e-185 < y < 1.55e-109

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 80.0

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.3%

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

   if 1.89999999999999995e150 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 31.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 31.1

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

   6. Applied rewrites 31.1%

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

Alternative 12: 51.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\left(t\_1 + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 -4e+203)
     (fma y i z)
     (if (<= t_2 2e+61)
       (+ (fma (log c) (- b 0.5) z) a)
       (if (<= t_2 1e+307) (+ (+ t_1 t) 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);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -4e+203) {
		tmp = fma(y, i, z);
	} else if (t_2 <= 2e+61) {
		tmp = fma(log(c), (b - 0.5), z) + a;
	} else if (t_2 <= 1e+307) {
		tmp = (t_1 + t) + 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(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -4e+203)
		tmp = fma(y, i, z);
	elseif (t_2 <= 2e+61)
		tmp = Float64(fma(log(c), Float64(b - 0.5), z) + a);
	elseif (t_2 <= 1e+307)
		tmp = Float64(Float64(t_1 + t) + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+203], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, 2e+61], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[(N[(t$95$1 + t), $MachinePrecision] + a), $MachinePrecision], N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e203

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -4e203 < (+.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)) < 1.9999999999999999e61

   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. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.0%

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

   if 1.9999999999999999e61 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999986e306

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \log y + t\right) + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \log y + t\right) + a \]

      2. lift-log.f64 52.4

         \[\leadsto \left(x \cdot \log y + t\right) + a \]

   7. Applied rewrites 52.4%

      \[\leadsto \left(x \cdot \log y + t\right) + a \]

   if 9.99999999999999986e306 < (+.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 13: 47.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\left(t\_1 + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 2e+18)
     (fma y i z)
     (if (<= t_2 1e+307) (+ (+ t_1 t) 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);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= 2e+18) {
		tmp = fma(y, i, z);
	} else if (t_2 <= 1e+307) {
		tmp = (t_1 + t) + 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(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= 2e+18)
		tmp = fma(y, i, z);
	elseif (t_2 <= 1e+307)
		tmp = Float64(Float64(t_1 + t) + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+18], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[(N[(t$95$1 + t), $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)) < 2e18

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999986e306

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \log y + t\right) + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \log y + t\right) + a \]

      2. lift-log.f64 52.3

         \[\leadsto \left(x \cdot \log y + t\right) + a \]

   7. Applied rewrites 52.3%

      \[\leadsto \left(x \cdot \log y + t\right) + a \]

   if 9.99999999999999986e306 < (+.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 14: 40.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;t\_1 + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 2e+18)
     (fma y i z)
     (if (<= t_2 1e+307) (+ t_1 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);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= 2e+18) {
		tmp = fma(y, i, z);
	} else if (t_2 <= 1e+307) {
		tmp = t_1 + 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(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= 2e+18)
		tmp = fma(y, i, z);
	elseif (t_2 <= 1e+307)
		tmp = Float64(t_1 + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+18], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[(t$95$1 + 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)) < 2e18

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999986e306

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-log.f64 35.4

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

   7. Applied rewrites 35.4%

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

   if 9.99999999999999986e306 < (+.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 92.6

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

   6. Applied rewrites 92.6%

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

Alternative 15: 40.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;t\_1 + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \log c \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 2e+18)
     (fma y i z)
     (if (<= t_2 1e+307) (+ t_1 a) (fma y i (* (log c) b))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= 2e+18) {
		tmp = fma(y, i, z);
	} else if (t_2 <= 1e+307) {
		tmp = t_1 + a;
	} else {
		tmp = fma(y, i, (log(c) * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= 2e+18)
		tmp = fma(y, i, z);
	elseif (t_2 <= 1e+307)
		tmp = Float64(t_1 + a);
	else
		tmp = fma(y, i, Float64(log(c) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+18], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[(t$95$1 + a), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * b), $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)) < 2e18

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999986e306

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-log.f64 35.4

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

   7. Applied rewrites 35.4%

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

   if 9.99999999999999986e306 < (+.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 91.0

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

   6. Applied rewrites 91.0%

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

Alternative 16: 40.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_1 + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 2e+18) (fma y i z) (if (<= t_2 5e+303) (+ t_1 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);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= 2e+18) {
		tmp = fma(y, i, z);
	} else if (t_2 <= 5e+303) {
		tmp = t_1 + a;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= 2e+18)
		tmp = fma(y, i, z);
	elseif (t_2 <= 5e+303)
		tmp = Float64(t_1 + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+18], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, 5e+303], N[(t$95$1 + a), $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)) < 2e18

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2e18 < (+.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)) < 4.9999999999999997e303

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-log.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y + a \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-log.f64 35.1

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

   7. Applied rewrites 35.1%

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

   if 4.9999999999999997e303 < (+.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 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in a around inf

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

      1. +-commutative 80.5

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

      2. +-commutative 80.5

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

      3. *-commutative 80.5

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

      4. +-commutative 80.5

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

      5. associate-+l+ 80.5

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

      6. *-commutative 80.5

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

   6. Applied rewrites 80.5%

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

Alternative 17: 39.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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))
      2e+18)
   (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)) <= 2e+18) {
		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)) <= 2e+18)
		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], 2e+18], 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)) < 2e18

   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. Taylor expanded in z around inf

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

   4. Applied rewrites 37.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2e18 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      8. lift-log.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c}\right) + y \cdot i \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right)} \cdot \log c\right) + y \cdot i \]

      11. lift-log.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \color{blue}{\log c}\right) + y \cdot i \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot i + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + \mathsf{fma}\left(\log y, x, z\right)\right)\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   5. Step-by-step derivation

      1. +-commutative 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      2. +-commutative 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      3. *-commutative 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      4. +-commutative 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      5. associate-+l+ 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      6. *-commutative 38.5

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   6. Applied rewrites 38.5%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 39.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -100.0) (+ z 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 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = z + 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 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -100.0)
		tmp = Float64(z + 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, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(z + a), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 96.0

         \[\leadsto i \cdot \color{blue}{y} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{i \cdot y} \]

   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)) < -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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\log c \cdot \left(b - \frac{1}{2}\right) + x \cdot \log y\right) + z\right) + t\right) + a \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      13. lift-log.f64 86.9

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a \]

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a} \]

   5. Taylor expanded in z around inf

      \[\leadsto z + a \]
   6. Step-by-step derivation

   7. Applied rewrites 34.4%

      \[\leadsto z + a \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + \color{blue}{y \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i} \]

      3. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} + y \cdot i \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right)} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot \log y + z\right) + t\right)} + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \log y + z\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      8. lift-log.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\log y} + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c}\right) + y \cdot i \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \color{blue}{\left(b - \frac{1}{2}\right)} \cdot \log c\right) + y \cdot i \]

      11. lift-log.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \color{blue}{\log c}\right) + y \cdot i \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot i + \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right)} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \left(a + t\right) + \mathsf{fma}\left(\log y, x, z\right)\right)\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   5. Step-by-step derivation

      1. +-commutative 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      2. +-commutative 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      3. *-commutative 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      4. +-commutative 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      5. associate-+l+ 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

      6. *-commutative 37.8

         \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   6. Applied rewrites 37.8%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 37.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+32}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 5.8e+32) (+ z a) (* i y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 5.8e+32) {
		tmp = z + a;
	} else {
		tmp = i * y;
	}
	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 (y <= 5.8d+32) then
        tmp = z + a
    else
        tmp = i * y
    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 (y <= 5.8e+32) {
		tmp = z + a;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= 5.8e+32:
		tmp = z + a
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 5.8e+32)
		tmp = Float64(z + a);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= 5.8e+32)
		tmp = z + a;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 5.8e+32], N[(z + a), $MachinePrecision], N[(i * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.80000000000000006e32

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\log c \cdot \left(b - \frac{1}{2}\right) + x \cdot \log y\right) + z\right) + t\right) + a \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      13. lift-log.f64 94.2

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a \]

   4. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a} \]

   5. Taylor expanded in z around inf

      \[\leadsto z + a \]
   6. Step-by-step derivation

   7. Applied rewrites 37.2%

      \[\leadsto z + a \]

   if 5.80000000000000006e32 < y

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 44.4

         \[\leadsto i \cdot \color{blue}{y} \]

   4. Applied rewrites 44.4%

      \[\leadsto \color{blue}{i \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -100:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \end{array} \]

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))
      -100.0)
   z
   (+ t a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0) {
		tmp = z;
	} else {
		tmp = t + 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)) <= (-100.0d0)) then
        tmp = z
    else
        tmp = t + 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)) <= -100.0) {
		tmp = z;
	} else {
		tmp = t + 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)) <= -100.0:
		tmp = z
	else:
		tmp = t + a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -100.0)
		tmp = z;
	else
		tmp = Float64(t + 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)) <= -100.0)
		tmp = z;
	else
		tmp = t + 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], -100.0], z, N[(t + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 16.0%

      \[\leadsto \color{blue}{z} \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\log c \cdot \left(b - \frac{1}{2}\right) + x \cdot \log y\right) + z\right) + t\right) + a \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      9. lift-log.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

      13. lift-log.f64 78.1

          \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a \]

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a} \]

   5. Taylor expanded in t around inf

      \[\leadsto t + a \]
   6. Step-by-step derivation

   7. Applied rewrites 31.1%

      \[\leadsto t + a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 23.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -100:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

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))
      -100.0)
   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)) <= -100.0) {
		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)) <= (-100.0d0)) 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)) <= -100.0) {
		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)) <= -100.0:
		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)) <= -100.0)
		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)) <= -100.0)
		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], -100.0], 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)) < -100

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 16.0%

      \[\leadsto \color{blue}{z} \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 16.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ z + a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (+ z a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return z + a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = z + a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return z + a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return z + a

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(z + a)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = z + a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z + a), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + \color{blue}{a} \]

   3. +-commutative N/A

      \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right) + a \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right) + t\right) + a \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\left(\log c \cdot \left(b - \frac{1}{2}\right) + x \cdot \log y\right) + z\right) + t\right) + a \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

   9. lift-log.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

   10. lift--.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, x \cdot \log y\right) + z\right) + t\right) + a \]

   11. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - \frac{1}{2}, \log y \cdot x\right) + z\right) + t\right) + a \]

   13. lift-log.f64 77.7

       \[\leadsto \left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a \]

4. Applied rewrites 77.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + z\right) + t\right) + a} \]

5. Taylor expanded in z around inf

   \[\leadsto z + a \]
6. Step-by-step derivation

7. Applied rewrites 30.9%

   \[\leadsto z + a \]
8. Add Preprocessing

Alternative 23: 16.4% accurate, 37.6× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 a)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return a

Julia

function code(x, y, z, t, a, b, c, i)
	return a
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
3. Step-by-step derivation

4. Applied rewrites 16.6%

   \[\leadsto \color{blue}{a} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(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)))