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

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + N[(a + N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision]), $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. 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 99.9%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 3: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := x \cdot \log y\\ \mathbf{if}\;\left(\left(\left(\left(t\_2 + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i \leq -120:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t + t\_2\right) + a\right) + t\_1\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))) (t_2 (* x (log y))))
   (if (<= (+ (+ (+ (+ (+ t_2 z) t) a) t_1) (* y i)) -120.0)
     (+ t (+ z (fma i y (fma x (log y) (* (log c) (- b 0.5))))))
     (+ (+ (+ (+ t t_2) a) t_1) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double t_2 = x * log(y);
	double tmp;
	if ((((((t_2 + z) + t) + a) + t_1) + (y * i)) <= -120.0) {
		tmp = t + (z + fma(i, y, fma(x, log(y), (log(c) * (b - 0.5)))));
	} else {
		tmp = (((t + t_2) + a) + t_1) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	t_2 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(t_2 + z) + t) + a) + t_1) + Float64(y * i)) <= -120.0)
		tmp = Float64(t + Float64(z + fma(i, y, fma(x, log(y), Float64(log(c) * Float64(b - 0.5))))));
	else
		tmp = Float64(Float64(Float64(Float64(t + t_2) + a) + t_1) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(t$95$2 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -120.0], N[(t + N[(z + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t + t$95$2), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -120

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if -120 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 84.4

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

   4. Applied rewrites 84.4%

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

Alternative 4: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+58}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 4e+58)
   (+ t (+ z (fma i y (fma x (log y) (* (log c) (- b 0.5))))))
   (+ (fma (- b 0.5) (log c) (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 4e+58) {
		tmp = t + (z + fma(i, y, fma(x, log(y), (log(c) * (b - 0.5)))));
	} else {
		tmp = fma((b - 0.5), log(c), fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 4e+58)
		tmp = Float64(t + Float64(z + fma(i, y, fma(x, log(y), Float64(log(c) * Float64(b - 0.5))))));
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 4e+58], N[(t + N[(z + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.99999999999999978e58

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if 3.99999999999999978e58 < a

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

Alternative 5: 90.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, a + \left(t + \left(z + x \cdot \log y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.15e-29)
   (fma (- b 0.5) (log c) (+ a (+ t (+ z (* x (log y))))))
   (+ (fma (- b 0.5) (log c) (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.15e-29) {
		tmp = fma((b - 0.5), log(c), (a + (t + (z + (x * log(y))))));
	} else {
		tmp = fma((b - 0.5), log(c), fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.15e-29)
		tmp = fma(Float64(b - 0.5), log(c), Float64(a + Float64(t + Float64(z + Float64(x * log(y))))));
	else
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.15e-29], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(a + N[(t + N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999996e-29

   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 99.9%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(b - \frac{1}{2}, \log c, \color{blue}{a + \left(t + \left(z + x \cdot \log y\right)\right)}\right) \]
   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 77.1

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

   6. Applied rewrites 77.1%

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

   if 1.14999999999999996e-29 < 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

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

      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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ t (fma i y (fma x (log y) (* (log c) (- b 0.5)))))))
   (if (<= x -2.15e+197)
     t_1
     (if (<= x 1.95e+229)
       (+ (fma (- b 0.5) (log c) (fma y i (+ z a))) t)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t + fma(i, y, fma(x, log(y), (log(c) * (b - 0.5))));
	double tmp;
	if (x <= -2.15e+197) {
		tmp = t_1;
	} else if (x <= 1.95e+229) {
		tmp = fma((b - 0.5), log(c), fma(y, i, (z + a))) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t + fma(i, y, fma(x, log(y), Float64(log(c) * Float64(b - 0.5)))))
	tmp = 0.0
	if (x <= -2.15e+197)
		tmp = t_1;
	elseif (x <= 1.95e+229)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, Float64(z + a))) + t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e+197], t$95$1, If[LessEqual[x, 1.95e+229], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.14999999999999998e197 or 1.9499999999999999e229 < 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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 70.4

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

   7. Applied rewrites 70.4%

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

   if -2.14999999999999998e197 < x < 1.9499999999999999e229

   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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

Alternative 7: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\log y\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+198}:\\ \;\;\;\;\left(t\_1 - \frac{y}{x} \cdot i\right) \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \frac{y \cdot i}{x}\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- (log y))))
   (if (<= x -1.05e+198)
     (* (- t_1 (* (/ y x) i)) (- x))
     (if (<= x 4.6e+229)
       (+ (fma (- b 0.5) (log c) (fma y i (+ z a))) t)
       (* (- t_1 (/ (* y i) x)) (- x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -log(y);
	double tmp;
	if (x <= -1.05e+198) {
		tmp = (t_1 - ((y / x) * i)) * -x;
	} else if (x <= 4.6e+229) {
		tmp = fma((b - 0.5), log(c), fma(y, i, (z + a))) + t;
	} else {
		tmp = (t_1 - ((y * i) / x)) * -x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-log(y))
	tmp = 0.0
	if (x <= -1.05e+198)
		tmp = Float64(Float64(t_1 - Float64(Float64(y / x) * i)) * Float64(-x));
	elseif (x <= 4.6e+229)
		tmp = Float64(fma(Float64(b - 0.5), log(c), fma(y, i, Float64(z + a))) + t);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(y * i) / x)) * Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = (-N[Log[y], $MachinePrecision])}, If[LessEqual[x, -1.05e+198], N[(N[(t$95$1 - N[(N[(y / x), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[x, 4.6e+229], N[(N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], N[(N[(t$95$1 - N[(N[(y * i), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000006e198

   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 x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.7

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

   7. Applied rewrites 35.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 35.3

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

   9. Applied rewrites 35.3%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \mathsf{fma}\left(-1, \log y, -1 \cdot \left(\frac{y}{x} \cdot i\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \mathsf{fma}\left(-1, \log y, -1 \cdot \left(\frac{y}{x} \cdot i\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(-1, \log y, -1 \cdot \left(\frac{y}{x} \cdot i\right)\right) \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1, \log y, -1 \cdot \left(\frac{y}{x} \cdot i\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \log y, -1 \cdot \left(\frac{y}{x} \cdot i\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   11. Applied rewrites 35.3%

       \[\leadsto \left(\left(-\log y\right) - \frac{y}{x} \cdot i\right) \cdot \color{blue}{\left(-x\right)} \]

   if -1.05000000000000006e198 < x < 4.5999999999999999e229

   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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

   if 4.5999999999999999e229 < 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 x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.7

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

   7. Applied rewrites 35.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \mathsf{fma}\left(-1, \log y, -1 \cdot \frac{i \cdot y}{x}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \mathsf{fma}\left(-1, \log y, -1 \cdot \frac{i \cdot y}{x}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(-1, \log y, -1 \cdot \frac{i \cdot y}{x}\right) \cdot x\right) \]

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

         \[\leadsto \mathsf{fma}\left(-1, \log y, -1 \cdot \frac{i \cdot y}{x}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \log y, -1 \cdot \frac{i \cdot y}{x}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   9. Applied rewrites 35.7%

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

Alternative 8: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, \log c \cdot b\right) + \left(t + z\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (fma i y (* (log c) b)) (+ t z))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -5e+190)
     t_1
     (if (<= t_2 1e+164) (+ (fma -0.5 (log c) (fma y i (+ z a))) t) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, y, (log(c) * b)) + (t + z);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -5e+190) {
		tmp = t_1;
	} else if (t_2 <= 1e+164) {
		tmp = fma(-0.5, log(c), fma(y, i, (z + a))) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(i, y, Float64(log(c) * b)) + Float64(t + z))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -5e+190)
		tmp = t_1;
	elseif (t_2 <= 1e+164)
		tmp = Float64(fma(-0.5, log(c), fma(y, i, Float64(z + a))) + t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(t + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+190], t$95$1, If[LessEqual[t$95$2, 1e+164], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -5.00000000000000036e190 or 1e164 < (*.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. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in b around inf

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

      1. lower-log.f64 67.9

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

   10. Applied rewrites 67.9%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

   12. Applied rewrites 67.9%

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

   if -5.00000000000000036e190 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 1e164

   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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.3%

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

Alternative 9: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.05 \cdot 10^{+61}:\\ \;\;\;\;t + \left(z + \mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(y, i, z + a\right)\right) + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 5.05e+61)
   (+ t (+ z (fma i y (* (log c) (- b 0.5)))))
   (+ (fma -0.5 (log c) (fma y i (+ z a))) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 5.05e+61) {
		tmp = t + (z + fma(i, y, (log(c) * (b - 0.5))));
	} else {
		tmp = fma(-0.5, log(c), fma(y, i, (z + a))) + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 5.05e+61)
		tmp = Float64(t + Float64(z + fma(i, y, Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = Float64(fma(-0.5, log(c), fma(y, i, Float64(z + a))) + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 5.05e+61], N[(t + N[(z + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(y * i + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0500000000000001e61

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if 5.0500000000000001e61 < a

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      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 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 83.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.3%

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

Alternative 10: 52.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ t_2 := \mathsf{fma}\left(i, y, \log c \cdot b\right) + \left(t + z\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+289}:\\ \;\;\;\;-\left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))
        (t_2 (+ (fma i y (* (log c) b)) (+ t z))))
   (if (<= t_1 5e+80) t_2 (if (<= t_1 1e+289) (- (- a)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double t_2 = fma(i, y, (log(c) * b)) + (t + z);
	double tmp;
	if (t_1 <= 5e+80) {
		tmp = t_2;
	} else if (t_1 <= 1e+289) {
		tmp = -(-a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	t_2 = Float64(fma(i, y, Float64(log(c) * b)) + Float64(t + z))
	tmp = 0.0
	if (t_1 <= 5e+80)
		tmp = t_2;
	elseif (t_1 <= 1e+289)
		tmp = Float64(-Float64(-a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(t + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+80], t$95$2, If[LessEqual[t$95$1, 1e+289], (-(-a)), t$95$2]]]]

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)) < 4.99999999999999961e80 or 1.0000000000000001e289 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in b around inf

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

      1. lower-log.f64 67.9

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

   10. Applied rewrites 67.9%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

   12. Applied rewrites 67.9%

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

   if 4.99999999999999961e80 < (+.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)) < 1.0000000000000001e289

   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 x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 15.3

         \[\leadsto -1 \cdot \left(-1 \cdot a\right) \]

   7. Applied rewrites 15.3%

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 \cdot a\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot a\right) \]

      3. lower-neg.f64 15.3

         \[\leadsto --1 \cdot a \]

      4. lift-*.f64 N/A

         \[\leadsto --1 \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(a\right)\right) \]

      6. lower-neg.f64 15.3

         \[\leadsto -\left(-a\right) \]

   9. Applied rewrites 15.3%

      \[\leadsto -\left(-a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 28.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;-1 \cdot \left(-1 \cdot z\right)\\ \mathbf{elif}\;t\_1 \leq 6 \cdot 10^{+307}:\\ \;\;\;\;-\left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* i y)
     (if (<= t_1 -100.0)
       (* -1.0 (* -1.0 z))
       (if (<= t_1 6e+307) (- (- a)) (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = -1.0 * (-1.0 * z);
	} else if (t_1 <= 6e+307) {
		tmp = -(-a);
	} else {
		tmp = i * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = -1.0 * (-1.0 * z);
	} else if (t_1 <= 6e+307) {
		tmp = -(-a);
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = i * y
	elif t_1 <= -100.0:
		tmp = -1.0 * (-1.0 * z)
	elif t_1 <= 6e+307:
		tmp = -(-a)
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -100.0)
		tmp = Float64(-1.0 * Float64(-1.0 * z));
	elseif (t_1 <= 6e+307)
		tmp = Float64(-Float64(-a));
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = i * y;
	elseif (t_1 <= -100.0)
		tmp = -1.0 * (-1.0 * z);
	elseif (t_1 <= 6e+307)
		tmp = -(-a);
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(-1.0 * N[(-1.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6e+307], (-(-a)), N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0 or 5.9999999999999997e307 < (+.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

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

      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 99.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 24.1

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

   6. Applied rewrites 24.1%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.8%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{z}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 16.6

         \[\leadsto -1 \cdot \left(-1 \cdot z\right) \]

   7. Applied rewrites 16.6%

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{z}\right) \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5.9999999999999997e307

   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 x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 15.3

         \[\leadsto -1 \cdot \left(-1 \cdot a\right) \]

   7. Applied rewrites 15.3%

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 \cdot a\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot a\right) \]

      3. lower-neg.f64 15.3

         \[\leadsto --1 \cdot a \]

      4. lift-*.f64 N/A

         \[\leadsto --1 \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(a\right)\right) \]

      6. lower-neg.f64 15.3

         \[\leadsto -\left(-a\right) \]

   9. Applied rewrites 15.3%

      \[\leadsto -\left(-a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 26.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+33}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq 6 \cdot 10^{+307}:\\ \;\;\;\;-\left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 2e+33) (* i y) (if (<= t_1 6e+307) (- (- a)) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= 2e+33) {
		tmp = i * y;
	} else if (t_1 <= 6e+307) {
		tmp = -(-a);
	} else {
		tmp = i * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= 2d+33) then
        tmp = i * y
    else if (t_1 <= 6d+307) then
        tmp = -(-a)
    else
        tmp = i * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= 2e+33) {
		tmp = i * y;
	} else if (t_1 <= 6e+307) {
		tmp = -(-a);
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= 2e+33:
		tmp = i * y
	elif t_1 <= 6e+307:
		tmp = -(-a)
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= 2e+33)
		tmp = Float64(i * y);
	elseif (t_1 <= 6e+307)
		tmp = Float64(-Float64(-a));
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= 2e+33)
		tmp = i * y;
	elseif (t_1 <= 6e+307)
		tmp = -(-a);
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+33], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, 6e+307], (-(-a)), N[(i * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 1.9999999999999999e33 or 5.9999999999999997e307 < (+.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

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

      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 99.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 24.1

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

   6. Applied rewrites 24.1%

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

   if 1.9999999999999999e33 < (+.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)) < 5.9999999999999997e307

   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 x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 15.3

         \[\leadsto -1 \cdot \left(-1 \cdot a\right) \]

   7. Applied rewrites 15.3%

      \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 \cdot a\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot a\right) \]

      3. lower-neg.f64 15.3

         \[\leadsto --1 \cdot a \]

      4. lift-*.f64 N/A

         \[\leadsto --1 \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(a\right)\right) \]

      6. lower-neg.f64 15.3

         \[\leadsto -\left(-a\right) \]

   9. Applied rewrites 15.3%

      \[\leadsto -\left(-a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 15.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ -\left(-a\right) \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := (-(-a))

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 74.2%

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

5. Taylor expanded in a around inf

   \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 15.3

      \[\leadsto -1 \cdot \left(-1 \cdot a\right) \]

7. Applied rewrites 15.3%

   \[\leadsto -1 \cdot \left(-1 \cdot \color{blue}{a}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(-1 \cdot a\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(-1 \cdot a\right) \]

   3. lower-neg.f64 15.3

      \[\leadsto --1 \cdot a \]

   4. lift-*.f64 N/A

      \[\leadsto --1 \cdot a \]

   5. mul-1-neg N/A

      \[\leadsto -\left(\mathsf{neg}\left(a\right)\right) \]

   6. lower-neg.f64 15.3

      \[\leadsto -\left(-a\right) \]

9. Applied rewrites 15.3%

   \[\leadsto -\left(-a\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))