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

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 94.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 84.3

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

4. Applied rewrites 84.3%

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

Alternative 3: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, -0.5, \mathsf{fma}\left(\log y, x, z\right)\right) + a\right) + t\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (+ (+ (fma (log c) -0.5 (fma (log y) x z)) a) t))))
   (if (<= x -9e+101)
     t_1
     (if (<= x 2.1e+33) (fma y i (+ (+ (fma (log c) (- b 0.5) z) a) t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, ((fma(log(c), -0.5, fma(log(y), x, z)) + a) + t));
	double tmp;
	if (x <= -9e+101) {
		tmp = t_1;
	} else if (x <= 2.1e+33) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + a) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(Float64(fma(log(c), -0.5, fma(log(y), x, z)) + a) + t))
	tmp = 0.0
	if (x <= -9e+101)
		tmp = t_1;
	elseif (x <= 2.1e+33)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + a) + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * -0.5 + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+101], t$95$1, If[LessEqual[x, 2.1e+33], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000004e101 or 2.1000000000000001e33 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in b around 0

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

   6. Applied rewrites 84.3%

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

   if -9.0000000000000004e101 < x < 2.1000000000000001e33

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.7%

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

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot -0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (fma i y (fma x (log y) (* (log c) -0.5)))))))
   (if (<= x -1.16e+104)
     t_1
     (if (<= x 2.2e+69) (fma y i (+ (+ (fma (log c) (- b 0.5) z) a) t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (z + fma(i, y, fma(x, log(y), (log(c) * -0.5))));
	double tmp;
	if (x <= -1.16e+104) {
		tmp = t_1;
	} else if (x <= 2.2e+69) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + a) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(z + fma(i, y, fma(x, log(y), Float64(log(c) * -0.5)))))
	tmp = 0.0
	if (x <= -1.16e+104)
		tmp = t_1;
	elseif (x <= 2.2e+69)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + a) + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(z + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.16e+104], t$95$1, If[LessEqual[x, 2.2e+69], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1599999999999999e104 or 2.2000000000000002e69 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 69.1%

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

   if -1.1599999999999999e104 < x < 2.2000000000000002e69

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.7%

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

Alternative 5: 84.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -3.4e+116)
   (fma y i (+ (+ (fma (log c) (- b 0.5) z) a) t))
   (+ a (fma i y (fma x (log y) (* (log c) (- b 0.5)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -3.4e+116) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + a) + t));
	} else {
		tmp = a + fma(i, y, fma(x, log(y), (log(c) * (b - 0.5))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -3.4e+116)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + a) + t));
	else
		tmp = Float64(a + fma(i, y, fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -3.4e+116], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(a + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000023e116

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.7%

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

   if -3.40000000000000023e116 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 69.1

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

   7. Applied rewrites 69.1%

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

Alternative 6: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+91}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(i, y, b \cdot \log c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 7.2e+91)
   (+ a (+ z (fma x (log y) (* (log c) (- b 0.5)))))
   (+ a (+ z (fma i y (* b (log c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 7.2e+91) {
		tmp = a + (z + fma(x, log(y), (log(c) * (b - 0.5))));
	} else {
		tmp = a + (z + fma(i, y, (b * log(c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 7.2e+91)
		tmp = Float64(a + Float64(z + fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = Float64(a + Float64(z + fma(i, y, Float64(b * log(c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 7.2e+91], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(z + N[(i * y + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2e91

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 73.8

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

   7. Applied rewrites 73.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 73.8

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 73.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   10. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-log.f64 N/A

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

       6. lower--.f64 62.0

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

   12. Applied rewrites 62.0%

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

   if 7.2e91 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-log.f64 67.9

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

   10. Applied rewrites 67.9%

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

Alternative 7: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+221}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -7e+154)
     t_1
     (if (<= x 2.8e+221) (+ a (+ z (fma i y (* (log c) (- b 0.5))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -7e+154) {
		tmp = t_1;
	} else if (x <= 2.8e+221) {
		tmp = a + (z + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -7e+154)
		tmp = t_1;
	elseif (x <= 2.8e+221)
		tmp = Float64(a + Float64(z + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+154], t$95$1, If[LessEqual[x, 2.8e+221], N[(a + N[(z + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000041e154 or 2.79999999999999989e221 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.0

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

   6. Applied rewrites 16.0%

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

   if -7.00000000000000041e154 < x < 2.79999999999999989e221

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 69.5

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

   7. Applied rewrites 69.5%

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

Alternative 8: 72.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\right) + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 2.8e+221)
   (fma y i (+ (+ (fma (log c) (- b 0.5) z) a) t))
   (* x (log y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 2.8e+221) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), z) + a) + t));
	} else {
		tmp = x * log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 2.8e+221)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), z) + a) + t));
	else
		tmp = Float64(x * log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 2.8e+221], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999989e221

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.7%

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

   if 2.79999999999999989e221 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.0

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

   6. Applied rewrites 16.0%

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

Alternative 9: 70.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+221}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(i, y, b \cdot \log c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -7e+154)
     t_1
     (if (<= x 2.8e+221) (+ a (+ z (fma i y (* b (log c))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -7e+154) {
		tmp = t_1;
	} else if (x <= 2.8e+221) {
		tmp = a + (z + fma(i, y, (b * log(c))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -7e+154)
		tmp = t_1;
	elseif (x <= 2.8e+221)
		tmp = Float64(a + Float64(z + fma(i, y, Float64(b * log(c)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+154], t$95$1, If[LessEqual[x, 2.8e+221], N[(a + N[(z + N[(i * y + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000041e154 or 2.79999999999999989e221 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.0

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

   6. Applied rewrites 16.0%

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

   if -7.00000000000000041e154 < x < 2.79999999999999989e221

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-log.f64 67.9

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

   10. Applied rewrites 67.9%

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

Alternative 10: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{z}{a} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   (fma y i (* (/ z a) a))
   (fma y i (* 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0) {
		tmp = fma(y, i, ((z / a) * a));
	} else {
		tmp = fma(y, i, (1.0 * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -50.0)
		tmp = fma(y, i, Float64(Float64(z / a) * a));
	else
		tmp = fma(y, i, Float64(1.0 * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], N[(y * i + N[(N[(z / a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

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

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 38.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 38.4

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

   8. Applied rewrites 38.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.4

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

   11. Applied rewrites 31.4%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

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

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 38.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 38.4

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

   8. Applied rewrites 38.4%

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

Alternative 11: 38.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.1e+154) t_1 (if (<= x 5.4e+165) (fma y i (* 1.0 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 = x * log(y);
	double tmp;
	if (x <= -5.1e+154) {
		tmp = t_1;
	} else if (x <= 5.4e+165) {
		tmp = fma(y, i, (1.0 * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5.1e+154)
		tmp = t_1;
	elseif (x <= 5.4e+165)
		tmp = fma(y, i, Float64(1.0 * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.1e+154], t$95$1, If[LessEqual[x, 5.4e+165], N[(y * i + N[(1.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0999999999999999e154 or 5.3999999999999999e165 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.0

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

   6. Applied rewrites 16.0%

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

   if -5.0999999999999999e154 < x < 5.3999999999999999e165

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

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

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 38.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 38.4

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

   8. Applied rewrites 38.4%

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

Alternative 12: 34.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, 1 \cdot a\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(1.0 * 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

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. sum-to-mult N/A

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

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

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

3. Applied rewrites 73.0%

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

4. Taylor expanded in a around inf

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

6. Applied rewrites 38.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 38.4

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

8. Applied rewrites 38.4%

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

Alternative 13: 16.5% accurate, 37.6× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

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

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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := t

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 84.3

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

4. Applied rewrites 84.3%

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

5. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 73.8

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

7. Applied rewrites 73.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 73.8

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 73.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 73.8

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

9. Applied rewrites 73.8%

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

10. Taylor expanded in t around inf

    \[\leadsto \color{blue}{t} \]
11. Step-by-step derivation

12. Applied rewrites 16.5%

    \[\leadsto \color{blue}{t} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(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)))