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

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

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

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

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, 0.9× speedup

Math

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

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 (fmin z a))) (fmax 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, fmin(z, a))) + fmax(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, fmin(z, a))) + fmax(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 + N[Min[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Max[z, a], $MachinePrecision]), $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: 99.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+
  (fmax (fmax z t) (fmax (fmin z t) a))
  (+ (fmin (fmin z 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) {
	return fmax(fmax(z, t), fmax(fmin(z, t), a)) + (fmin(fmin(z, t), a) + 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(fmax(fmax(z, t), fmax(fmin(z, t), a)) + Float64(fmin(fmin(z, t), a) + 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[(N[Max[N[Max[z, t], $MachinePrecision], N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]], $MachinePrecision] + N[(N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision] + 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 85.3%

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

4. Applied rewrites 85.3%

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

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_5 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, t\_2 + \left(t\_3 + \left(t\_4 + t\_5\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, t\_4\right) + t\_2\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_4 + \mathsf{fma}\left(i, y, 1 \cdot t\_5\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmax (fmin z t) a))
        (t_2 (fmax (fmax z t) t_1))
        (t_3 (fmin (fmax z t) t_1))
        (t_4 (fmin (fmin z t) a))
        (t_5 (* x (log y))))
   (if (<= x -2.15e+125)
     (fma (- b 0.5) (log c) (+ t_2 (+ t_3 (+ t_4 t_5))))
     (if (<= x 1.15e+203)
       (fma y i (+ (+ (fma (log c) (- b 0.5) t_4) t_2) t_3))
       (+ t_2 (+ t_4 (fma i y (* 1.0 t_5))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmax(fmin(z, t), a);
	double t_2 = fmax(fmax(z, t), t_1);
	double t_3 = fmin(fmax(z, t), t_1);
	double t_4 = fmin(fmin(z, t), a);
	double t_5 = x * log(y);
	double tmp;
	if (x <= -2.15e+125) {
		tmp = fma((b - 0.5), log(c), (t_2 + (t_3 + (t_4 + t_5))));
	} else if (x <= 1.15e+203) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), t_4) + t_2) + t_3));
	} else {
		tmp = t_2 + (t_4 + fma(i, y, (1.0 * t_5)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmax(fmin(z, t), a)
	t_2 = fmax(fmax(z, t), t_1)
	t_3 = fmin(fmax(z, t), t_1)
	t_4 = fmin(fmin(z, t), a)
	t_5 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.15e+125)
		tmp = fma(Float64(b - 0.5), log(c), Float64(t_2 + Float64(t_3 + Float64(t_4 + t_5))));
	elseif (x <= 1.15e+203)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), t_4) + t_2) + t_3));
	else
		tmp = Float64(t_2 + Float64(t_4 + fma(i, y, Float64(1.0 * t_5))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$5 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e+125], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(t$95$2 + N[(t$95$3 + N[(t$95$4 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e+203], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$4 + N[(i * y + N[(1.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.15000000000000018e125

   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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 76.9%

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

   6. Applied rewrites 76.9%

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

   if -2.15000000000000018e125 < x < 1.15e203

   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

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

   6. Applied rewrites 83.5%

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

   if 1.15e203 < 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 85.3%

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

   4. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

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

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

      5. lower-unsound--.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.0%

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 68.4%

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

Alternative 4: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_1\right)\\ t_4 := t\_3 + \left(t\_2 + \mathsf{fma}\left(i, y, 1 \cdot \left(x \cdot \log y\right)\right)\right)\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, t\_2\right) + t\_3\right) + \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmax (fmin z t) a))
        (t_2 (fmin (fmin z t) a))
        (t_3 (fmax (fmax z t) t_1))
        (t_4 (+ t_3 (+ t_2 (fma i y (* 1.0 (* x (log y))))))))
   (if (<= x -9.2e+58)
     t_4
     (if (<= x 1.15e+203)
       (fma y i (+ (+ (fma (log c) (- b 0.5) t_2) t_3) (fmin (fmax z t) t_1)))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmax(fmin(z, t), a);
	double t_2 = fmin(fmin(z, t), a);
	double t_3 = fmax(fmax(z, t), t_1);
	double t_4 = t_3 + (t_2 + fma(i, y, (1.0 * (x * log(y)))));
	double tmp;
	if (x <= -9.2e+58) {
		tmp = t_4;
	} else if (x <= 1.15e+203) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), t_2) + t_3) + fmin(fmax(z, t), t_1)));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmax(fmin(z, t), a)
	t_2 = fmin(fmin(z, t), a)
	t_3 = fmax(fmax(z, t), t_1)
	t_4 = Float64(t_3 + Float64(t_2 + fma(i, y, Float64(1.0 * Float64(x * log(y))))))
	tmp = 0.0
	if (x <= -9.2e+58)
		tmp = t_4;
	elseif (x <= 1.15e+203)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), t_2) + t_3) + fmin(fmax(z, t), t_1)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(t$95$2 + N[(i * y + N[(1.0 * N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+58], t$95$4, If[LessEqual[x, 1.15e+203], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] + N[Min[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000001e58 or 1.15e203 < 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 85.3%

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

   4. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

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

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

      5. lower-unsound--.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.0%

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 68.4%

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

   if -9.2000000000000001e58 < x < 1.15e203

   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

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

   6. Applied rewrites 83.5%

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

Alternative 5: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\right)\\ t_3 := t\_2 + \left(t\_1 + \mathsf{fma}\left(i, y, 1 \cdot \left(x \cdot \log y\right)\right)\right)\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;t\_2 + \left(t\_1 + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmax z t) (fmax (fmin z t) a)))
        (t_3 (+ t_2 (+ t_1 (fma i y (* 1.0 (* x (log y))))))))
   (if (<= x -9.2e+58)
     t_3
     (if (<= x 1.15e+203)
       (+ t_2 (+ t_1 (fma i y (* (log c) (- b 0.5)))))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmax(z, t), fmax(fmin(z, t), a));
	double t_3 = t_2 + (t_1 + fma(i, y, (1.0 * (x * log(y)))));
	double tmp;
	if (x <= -9.2e+58) {
		tmp = t_3;
	} else if (x <= 1.15e+203) {
		tmp = t_2 + (t_1 + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmax(z, t), fmax(fmin(z, t), a))
	t_3 = Float64(t_2 + Float64(t_1 + fma(i, y, Float64(1.0 * Float64(x * log(y))))))
	tmp = 0.0
	if (x <= -9.2e+58)
		tmp = t_3;
	elseif (x <= 1.15e+203)
		tmp = Float64(t_2 + Float64(t_1 + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[z, t], $MachinePrecision], N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(t$95$1 + N[(i * y + N[(1.0 * N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+58], t$95$3, If[LessEqual[x, 1.15e+203], N[(t$95$2 + N[(t$95$1 + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000001e58 or 1.15e203 < 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 85.3%

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

   4. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

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

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

      5. lower-unsound--.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.0%

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 68.4%

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

   if -9.2000000000000001e58 < x < 1.15e203

   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 85.3%

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

   4. Applied rewrites 85.3%

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

   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 69.3%

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

   7. Applied rewrites 69.3%

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

Alternative 6: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;t\_1 \leq -4.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \left(\mathsf{fma}\left(\log c, b - 0.5, t\_1\right) + t\_3\right) + \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmax (fmax z t) t_2)))
   (if (<= t_1 -4.1e+88)
     (fma y i (+ (+ (fma (log c) (- b 0.5) t_1) t_3) (fmin (fmax z t) t_2)))
     (+ t_3 (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 t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmax(fmax(z, t), t_2);
	double tmp;
	if (t_1 <= -4.1e+88) {
		tmp = fma(y, i, ((fma(log(c), (b - 0.5), t_1) + t_3) + fmin(fmax(z, t), t_2)));
	} else {
		tmp = t_3 + 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)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmax(fmax(z, t), t_2)
	tmp = 0.0
	if (t_1 <= -4.1e+88)
		tmp = fma(y, i, Float64(Float64(fma(log(c), Float64(b - 0.5), t_1) + t_3) + fmin(fmax(z, t), t_2)));
	else
		tmp = Float64(t_3 + 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_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[t$95$1, -4.1e+88], N[(y * i + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + 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 < -4.10000000000000028e88

   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

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

   6. Applied rewrites 83.5%

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

   if -4.10000000000000028e88 < 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 85.3%

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

   4. Applied rewrites 85.3%

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

   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 70.6%

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

   7. Applied rewrites 70.6%

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

Alternative 7: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right) + \left(\mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right) + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_3 \cdot t\_1}{t\_3}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2)))
   (if (<= x -3.2e+254)
     t_1
     (if (<= x 2.4e+258)
       (+
        (fmax (fmax z t) t_2)
        (+ (fmin (fmin z t) a) (fma i y (* (log c) (- b 0.5)))))
       (fma y i (/ (* t_3 t_1) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double tmp;
	if (x <= -3.2e+254) {
		tmp = t_1;
	} else if (x <= 2.4e+258) {
		tmp = fmax(fmax(z, t), t_2) + (fmin(fmin(z, t), a) + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = fma(y, i, ((t_3 * t_1) / t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	tmp = 0.0
	if (x <= -3.2e+254)
		tmp = t_1;
	elseif (x <= 2.4e+258)
		tmp = Float64(fmax(fmax(z, t), t_2) + Float64(fmin(fmin(z, t), a) + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = fma(y, i, Float64(Float64(t_3 * t_1) / t_3));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[x, -3.2e+254], t$95$1, If[LessEqual[x, 2.4e+258], N[(N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision] + N[(N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision] + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(t$95$3 * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999998e254

   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 17.3%

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

   6. Applied rewrites 17.3%

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

   if -3.1999999999999998e254 < x < 2.4e258

   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 85.3%

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

   4. Applied rewrites 85.3%

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

   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 69.3%

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

   7. Applied rewrites 69.3%

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

   if 2.4e258 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 34.8%

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

   8. Applied rewrites 34.8%

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

Alternative 8: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{fma}\left(y, i, \frac{b \cdot \left(t\_3 \cdot \log c\right)}{t\_3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+182}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(-0.5, \log c, \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right) + \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c)))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fma y i (/ (* b (* t_3 (log c))) t_3))))
   (if (<= t_1 -1e+182)
     t_4
     (if (<= t_1 1e+212)
       (fma
        y
        i
        (+
         (fma -0.5 (log c) (+ (fmin (fmin z t) a) (fmax (fmax z t) t_2)))
         t_3))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fma(y, i, ((b * (t_3 * log(c))) / t_3));
	double tmp;
	if (t_1 <= -1e+182) {
		tmp = t_4;
	} else if (t_1 <= 1e+212) {
		tmp = fma(y, i, (fma(-0.5, log(c), (fmin(fmin(z, t), a) + fmax(fmax(z, t), t_2))) + t_3));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fma(y, i, Float64(Float64(b * Float64(t_3 * log(c))) / t_3))
	tmp = 0.0
	if (t_1 <= -1e+182)
		tmp = t_4;
	elseif (t_1 <= 1e+212)
		tmp = fma(y, i, Float64(fma(-0.5, log(c), Float64(fmin(fmin(z, t), a) + fmax(fmax(z, t), t_2))) + t_3));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(y * i + N[(N[(b * N[(t$95$3 * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+182], t$95$4, If[LessEqual[t$95$1, 1e+212], N[(y * i + N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision] + N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1.0000000000000001e182 or 9.9999999999999991e211 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 34.2%

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

   8. Applied rewrites 34.2%

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

   if -1.0000000000000001e182 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999991e211

   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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.4%

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

Alternative 9: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{fma}\left(y, i, \frac{b \cdot \left(t\_3 \cdot \log c\right)}{t\_3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+182}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(i, y, \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right) + \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c)))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fma y i (/ (* b (* t_3 (log c))) t_3))))
   (if (<= t_1 -1e+182)
     t_4
     (if (<= t_1 1e+212)
       (fma
        -0.5
        (log c)
        (+ (fma i y (+ (fmax (fmax z t) t_2) (fmin (fmin z t) a))) t_3))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fma(y, i, ((b * (t_3 * log(c))) / t_3));
	double tmp;
	if (t_1 <= -1e+182) {
		tmp = t_4;
	} else if (t_1 <= 1e+212) {
		tmp = fma(-0.5, log(c), (fma(i, y, (fmax(fmax(z, t), t_2) + fmin(fmin(z, t), a))) + t_3));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fma(y, i, Float64(Float64(b * Float64(t_3 * log(c))) / t_3))
	tmp = 0.0
	if (t_1 <= -1e+182)
		tmp = t_4;
	elseif (t_1 <= 1e+212)
		tmp = fma(-0.5, log(c), Float64(fma(i, y, Float64(fmax(fmax(z, t), t_2) + fmin(fmin(z, t), a))) + t_3));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(y * i + N[(N[(b * N[(t$95$3 * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+182], t$95$4, If[LessEqual[t$95$1, 1e+212], N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision] + N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1.0000000000000001e182 or 9.9999999999999991e211 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 34.2%

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

   8. Applied rewrites 34.2%

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

   if -1.0000000000000001e182 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999991e211

   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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. associate-+r+ N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.5%

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

   7. Taylor expanded in b around 0

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

   9. Applied rewrites 68.4%

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

Alternative 10: 54.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_5 := \left(\left(\left(\left(x \cdot \log y + t\_1\right) + t\_3\right) + t\_4\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_1}{t\_3} \cdot t\_3\right)\\ \mathbf{elif}\;t\_5 \leq -100:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_3 \cdot t\_1}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_4 \cdot t\_3}{t\_3}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fmax (fmax z t) t_2))
        (t_5
         (+
          (+ (+ (+ (+ (* x (log y)) t_1) t_3) t_4) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_5 -5e+244)
     (fma y i (* (/ t_1 t_3) t_3))
     (if (<= t_5 -100.0)
       (fma y i (/ (* t_3 t_1) t_3))
       (fma y i (/ (* t_4 t_3) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fmax(fmax(z, t), t_2);
	double t_5 = (((((x * log(y)) + t_1) + t_3) + t_4) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_5 <= -5e+244) {
		tmp = fma(y, i, ((t_1 / t_3) * t_3));
	} else if (t_5 <= -100.0) {
		tmp = fma(y, i, ((t_3 * t_1) / t_3));
	} else {
		tmp = fma(y, i, ((t_4 * t_3) / t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fmax(fmax(z, t), t_2)
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + t_3) + t_4) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_5 <= -5e+244)
		tmp = fma(y, i, Float64(Float64(t_1 / t_3) * t_3));
	elseif (t_5 <= -100.0)
		tmp = fma(y, i, Float64(Float64(t_3 * t_1) / t_3));
	else
		tmp = fma(y, i, Float64(Float64(t_4 * t_3) / t_3));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -5e+244], N[(y * i + N[(N[(t$95$1 / t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -100.0], N[(y * i + N[(N[(t$95$3 * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(t$95$4 * t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.00000000000000022e244

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.6%

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

   11. Applied rewrites 31.6%

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

   if -5.00000000000000022e244 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 34.2%

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

   8. Applied rewrites 34.2%

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

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 34.1%

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

   8. Applied rewrites 34.1%

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

Alternative 11: 51.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + t\_1\right) + t\_3\right) + t\_4\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_1}{t\_3} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_4 \cdot t\_3}{t\_3}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fmax (fmax z t) t_2)))
   (if (<=
        (+
         (+ (+ (+ (+ (* x (log y)) t_1) t_3) t_4) (* (- b 0.5) (log c)))
         (* y i))
        -5e+21)
     (fma y i (* (/ t_1 t_3) t_3))
     (fma y i (/ (* t_4 t_3) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fmax(fmax(z, t), t_2);
	double tmp;
	if (((((((x * log(y)) + t_1) + t_3) + t_4) + ((b - 0.5) * log(c))) + (y * i)) <= -5e+21) {
		tmp = fma(y, i, ((t_1 / t_3) * t_3));
	} else {
		tmp = fma(y, i, ((t_4 * t_3) / t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fmax(fmax(z, t), t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + t_3) + t_4) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -5e+21)
		tmp = fma(y, i, Float64(Float64(t_1 / t_3) * t_3));
	else
		tmp = fma(y, i, Float64(Float64(t_4 * t_3) / t_3));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -5e+21], N[(y * i + N[(N[(t$95$1 / t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(t$95$4 * t$95$3), $MachinePrecision] / t$95$3), $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)) < -5e21

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.6%

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

   11. Applied rewrites 31.6%

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

   if -5e21 < (+.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. +-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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult-rev N/A

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

      4. add-to-fraction N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 34.1%

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

   8. Applied rewrites 34.1%

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

Alternative 12: 48.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + t\_1\right) + t\_3\right) + t\_4\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_1}{t\_3} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{t\_4}{t\_3} \cdot t\_3\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmin (fmin z t) a))
        (t_2 (fmax (fmin z t) a))
        (t_3 (fmin (fmax z t) t_2))
        (t_4 (fmax (fmax z t) t_2)))
   (if (<=
        (+
         (+ (+ (+ (+ (* x (log y)) t_1) t_3) t_4) (* (- b 0.5) (log c)))
         (* y i))
        -5e+21)
     (fma y i (* (/ t_1 t_3) t_3))
     (fma y i (* (/ t_4 t_3) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fmax(fmax(z, t), t_2);
	double tmp;
	if (((((((x * log(y)) + t_1) + t_3) + t_4) + ((b - 0.5) * log(c))) + (y * i)) <= -5e+21) {
		tmp = fma(y, i, ((t_1 / t_3) * t_3));
	} else {
		tmp = fma(y, i, ((t_4 / t_3) * t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fmax(fmax(z, t), t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + t_3) + t_4) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -5e+21)
		tmp = fma(y, i, Float64(Float64(t_1 / t_3) * t_3));
	else
		tmp = fma(y, i, Float64(Float64(t_4 / t_3) * t_3));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -5e+21], N[(y * i + N[(N[(t$95$1 / t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$3), $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)) < -5e21

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.6%

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

   11. Applied rewrites 31.6%

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

   if -5e21 < (+.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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 31.8%

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

   11. Applied rewrites 31.8%

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

Alternative 13: 40.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := x \cdot \log y\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_1\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+125}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \frac{\mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_1\right)}{t\_3} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fmax (fmin z t) a))
        (t_2 (* x (log y)))
        (t_3 (fmin (fmax z t) t_1)))
   (if (<= x -2.15e+125)
     t_2
     (if (<= x 2e+211) (fma y i (* (/ (fmax (fmax z t) t_1) t_3) t_3)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmax(fmin(z, t), a);
	double t_2 = x * log(y);
	double t_3 = fmin(fmax(z, t), t_1);
	double tmp;
	if (x <= -2.15e+125) {
		tmp = t_2;
	} else if (x <= 2e+211) {
		tmp = fma(y, i, ((fmax(fmax(z, t), t_1) / t_3) * t_3));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmax(fmin(z, t), a)
	t_2 = Float64(x * log(y))
	t_3 = fmin(fmax(z, t), t_1)
	tmp = 0.0
	if (x <= -2.15e+125)
		tmp = t_2;
	elseif (x <= 2e+211)
		tmp = fma(y, i, Float64(Float64(fmax(fmax(z, t), t_1) / t_3) * t_3));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[x, -2.15e+125], t$95$2, If[LessEqual[x, 2e+211], N[(y * i + N[(N[(N[Max[N[Max[z, t], $MachinePrecision], t$95$1], $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15000000000000018e125 or 1.9999999999999999e211 < 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 17.3%

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

   6. Applied rewrites 17.3%

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

   if -2.15000000000000018e125 < x < 1.9999999999999999e211

   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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 31.8%

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

   11. Applied rewrites 31.8%

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

Alternative 14: 38.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 6.1 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-101}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot \mathsf{min}\left(t, a\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= y 6.1e-153)
     t_1
     (if (<= y 9.6e-101)
       (* b (log c))
       (if (<= y 8e+34) t_1 (fma y i (* 1.0 (fmin t a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 6.1e-153) {
		tmp = t_1;
	} else if (y <= 9.6e-101) {
		tmp = b * log(c);
	} else if (y <= 8e+34) {
		tmp = t_1;
	} else {
		tmp = fma(y, i, (1.0 * fmin(t, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 6.1e-153)
		tmp = t_1;
	elseif (y <= 9.6e-101)
		tmp = Float64(b * log(c));
	elseif (y <= 8e+34)
		tmp = t_1;
	else
		tmp = fma(y, i, Float64(1.0 * fmin(t, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.1e-153], t$95$1, If[LessEqual[y, 9.6e-101], N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+34], t$95$1, N[(y * i + N[(1.0 * N[Min[t, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.10000000000000046e-153 or 9.6e-101 < y < 7.99999999999999956e34

   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 17.3%

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

   6. Applied rewrites 17.3%

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

   if 6.10000000000000046e-153 < y < 9.6e-101

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 16.5%

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

   4. Applied rewrites 16.5%

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

   if 7.99999999999999956e34 < 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. 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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

Alternative 15: 36.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, 1 \cdot \mathsf{min}\left(t, a\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 8e+34) (* x (log y)) (fma y i (* 1.0 (fmin t a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 8e+34) {
		tmp = x * log(y);
	} else {
		tmp = fma(y, i, (1.0 * fmin(t, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 8e+34)
		tmp = Float64(x * log(y));
	else
		tmp = fma(y, i, Float64(1.0 * fmin(t, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 8e+34], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(y * i + N[(1.0 * N[Min[t, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999956e34

   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 17.3%

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

   6. Applied rewrites 17.3%

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

   if 7.99999999999999956e34 < 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. 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. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 37.9%

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

   8. Applied rewrites 37.9%

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

Alternative 16: 35.7% accurate, 3.1× speedup

Math

\[\mathsf{fma}\left(y, i, 1 \cdot \mathsf{min}\left(t, a\right)\right) \]

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + N[(1.0 * N[Min[t, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   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. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. sum-to-mult N/A

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

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

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

3. Applied rewrites 72.9%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 37.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 37.9%

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

8. Applied rewrites 37.9%

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

Reproduce

herbie shell --seed 2025183 
(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)))