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

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 13

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

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

3. 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: 85.0% 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

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 84.5

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

4. Applied rewrites 84.5%

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

Alternative 3: 85.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   (+ (+ (fma i y (fma (log c) (- b 0.5) (* (log y) x))) z) t)
   (+ (+ (fma i y (fma (log y) x (* (log c) (- b 0.5)))) 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)) <= -50.0) {
		tmp = (fma(i, y, fma(log(c), (b - 0.5), (log(y) * x))) + z) + t;
	} else {
		tmp = (fma(i, y, fma(log(y), x, (log(c) * (b - 0.5)))) + 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)) <= -50.0)
		tmp = Float64(Float64(fma(i, y, fma(log(c), Float64(b - 0.5), Float64(log(y) * x))) + z) + t);
	else
		tmp = Float64(Float64(fma(i, y, fma(log(y), x, Float64(log(c) * Float64(b - 0.5)))) + t) + a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 84.8

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

   6. Applied rewrites 84.8%

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

Alternative 4: 84.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

Alternative 5: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\\ \mathbf{if}\;b \leq -6.8 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(x, \log y, z\right) + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (+ (fma (- b 0.5) (log c) z) a))))
   (if (<= b -6.8e+158)
     t_1
     (if (<= b 3.3e+160)
       (fma y i (fma -0.5 (log c) (+ (fma x (log y) z) a)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (fma((b - 0.5), log(c), z) + a));
	double tmp;
	if (b <= -6.8e+158) {
		tmp = t_1;
	} else if (b <= 3.3e+160) {
		tmp = fma(y, i, fma(-0.5, log(c), (fma(x, log(y), z) + a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(fma(Float64(b - 0.5), log(c), z) + a))
	tmp = 0.0
	if (b <= -6.8e+158)
		tmp = t_1;
	elseif (b <= 3.3e+160)
		tmp = fma(y, i, fma(-0.5, log(c), Float64(fma(x, log(y), z) + a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.8e+158], t$95$1, If[LessEqual[b, 3.3e+160], N[(y * i + N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.7999999999999998e158 or 3.2999999999999997e160 < b

   1. Initial program 99.8%

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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 69.3

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

   9. Applied rewrites 69.3%

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

   if -6.7999999999999998e158 < b < 3.2999999999999997e160

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 69.3

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-log.f64 N/A

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

   9. Applied rewrites 69.3%

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

Alternative 6: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right) + t\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 4.5e+230)
   (fma y i (+ (fma (- b 0.5) (log c) z) a))
   (+ (+ (fma (log c) (- b 0.5) (* (log y) x)) 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 <= 4.5e+230) {
		tmp = fma(y, i, (fma((b - 0.5), log(c), z) + a));
	} else {
		tmp = (fma(log(c), (b - 0.5), (log(y) * x)) + t) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 4.5e+230)
		tmp = fma(y, i, Float64(fma(Float64(b - 0.5), log(c), z) + a));
	else
		tmp = Float64(Float64(fma(log(c), Float64(b - 0.5), Float64(log(y) * x)) + t) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 4.5e+230], N[(y * i + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999999e230

   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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 69.3

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

   9. Applied rewrites 69.3%

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

   if 4.4999999999999999e230 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 61.9

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

   7. Applied rewrites 61.9%

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

Alternative 7: 71.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b - 0.5, \log c, z\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, -0.5, \mathsf{fma}\left(x, \log y, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 4.5e+230)
   (fma y i (+ (fma (- b 0.5) (log c) z) a))
   (+ (fma (log c) -0.5 (fma x (log y) 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 <= 4.5e+230) {
		tmp = fma(y, i, (fma((b - 0.5), log(c), z) + a));
	} else {
		tmp = fma(log(c), -0.5, fma(x, log(y), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 4.5e+230)
		tmp = fma(y, i, Float64(fma(Float64(b - 0.5), log(c), z) + a));
	else
		tmp = Float64(fma(log(c), -0.5, fma(x, log(y), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 4.5e+230], N[(y * i + N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * -0.5 + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999999e230

   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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 69.3

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

   9. Applied rewrites 69.3%

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

   if 4.4999999999999999e230 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 61.9

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

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 46.8%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. associate-+l+ N/A

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

       7. lift-*.f64 N/A

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

       8. lift-log.f64 N/A

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

       9. *-commutative N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lift-log.f64 N/A

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

       13. +-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lift-log.f64 46.8

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

   12. Applied rewrites 46.8%

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

Alternative 8: 69.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   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

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

3. 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 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

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 69.3

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

9. Applied rewrites 69.3%

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

Alternative 9: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \mathsf{fma}\left(b - 0.5, \log c, a\right)\right)\\ \mathbf{if}\;b - 0.5 \leq -1 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(-0.5, \log c, z\right) + a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (fma (- b 0.5) (log c) a))))
   (if (<= (- b 0.5) -1e+206)
     t_1
     (if (<= (- b 0.5) 5e+203) (fma y i (+ (fma -0.5 (log c) z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, fma((b - 0.5), log(c), a));
	double tmp;
	if ((b - 0.5) <= -1e+206) {
		tmp = t_1;
	} else if ((b - 0.5) <= 5e+203) {
		tmp = fma(y, i, (fma(-0.5, log(c), z) + a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, fma(Float64(b - 0.5), log(c), a))
	tmp = 0.0
	if (Float64(b - 0.5) <= -1e+206)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 5e+203)
		tmp = fma(y, i, Float64(fma(-0.5, log(c), z) + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -1e+206], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 5e+203], N[(y * i + N[(N[(-0.5 * N[Log[c], $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -1e206 or 4.99999999999999994e203 < (-.f64 b #s(literal 1/2 binary64))

   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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 69.3

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

   9. Applied rewrites 69.3%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-log.f64 54.8

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

   12. Applied rewrites 54.8%

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

   if -1e206 < (-.f64 b #s(literal 1/2 binary64)) < 4.99999999999999994e203

   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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 69.3

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

   9. Applied rewrites 69.3%

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

   10. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-log.f64 54.4

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

   12. Applied rewrites 54.4%

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

Alternative 10: 58.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   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

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

3. 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 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

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 69.3

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

9. Applied rewrites 69.3%

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

10. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lift--.f64 N/A

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

    5. lift-log.f64 54.8

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

12. Applied rewrites 54.8%

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

Alternative 11: 54.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+307}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, t\right) + a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+307)
     (* i y)
     (if (<= t_1 1e+308) (+ (fma (- b 0.5) (log c) t) a) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+307) {
		tmp = i * y;
	} else if (t_1 <= 1e+308) {
		tmp = fma((b - 0.5), log(c), t) + a;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+307)
		tmp = Float64(i * y);
	elseif (t_1 <= 1e+308)
		tmp = Float64(fma(Float64(b - 0.5), log(c), t) + a);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+307], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision], N[(i * y), $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)) < -3.99999999999999994e307 or 1e308 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 24.8

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

   9. Applied rewrites 24.8%

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

   if -3.99999999999999994e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1e308

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 61.9

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

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 46.7

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

   10. Applied rewrites 46.7%

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

Alternative 12: 32.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ \mathbf{if}\;b - 0.5 \leq -8 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b - 0.5 \leq 5 \cdot 10^{+205}:\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b)))
   (if (<= (- b 0.5) -8e+158) t_1 (if (<= (- b 0.5) 5e+205) (* i y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double tmp;
	if ((b - 0.5) <= -8e+158) {
		tmp = t_1;
	} else if ((b - 0.5) <= 5e+205) {
		tmp = i * y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = log(c) * b
    if ((b - 0.5d0) <= (-8d+158)) then
        tmp = t_1
    else if ((b - 0.5d0) <= 5d+205) then
        tmp = i * y
    else
        tmp = t_1
    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 t_1 = Math.log(c) * b;
	double tmp;
	if ((b - 0.5) <= -8e+158) {
		tmp = t_1;
	} else if ((b - 0.5) <= 5e+205) {
		tmp = i * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = math.log(c) * b
	tmp = 0
	if (b - 0.5) <= -8e+158:
		tmp = t_1
	elif (b - 0.5) <= 5e+205:
		tmp = i * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	tmp = 0.0
	if (Float64(b - 0.5) <= -8e+158)
		tmp = t_1;
	elseif (Float64(b - 0.5) <= 5e+205)
		tmp = Float64(i * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = log(c) * b;
	tmp = 0.0;
	if ((b - 0.5) <= -8e+158)
		tmp = t_1;
	elseif ((b - 0.5) <= 5e+205)
		tmp = i * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -8e+158], t$95$1, If[LessEqual[N[(b - 0.5), $MachinePrecision], 5e+205], N[(i * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -7.99999999999999962e158 or 5.0000000000000002e205 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 16.6

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

   4. Applied rewrites 16.6%

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

   if -7.99999999999999962e158 < (-.f64 b #s(literal 1/2 binary64)) < 5.0000000000000002e205

   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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

   3. 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 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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.3

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 24.8

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

   9. Applied rewrites 24.8%

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

Alternative 13: 24.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ i \cdot y \end{array} \]

FPCore

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

C

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

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 = i * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return i * y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(i * y), $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

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

3. 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 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

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in y around inf

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

   1. lower-*.f64 24.8

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

9. Applied rewrites 24.8%

   \[\leadsto \color{blue}{i \cdot y} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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)))