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

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. 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. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 22.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* i y)
     (if (<= t_1 -2e+41) z (if (<= t_1 INFINITY) 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 <= -2e+41) {
		tmp = z;
	} else if (t_1 <= ((double) INFINITY)) {
		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 <= -2e+41) {
		tmp = z;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		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 <= -2e+41:
		tmp = z
	elif t_1 <= math.inf:
		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 <= -2e+41)
		tmp = z;
	elseif (t_1 <= Inf)
		tmp = 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 <= -2e+41)
		tmp = z;
	elseif (t_1 <= Inf)
		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, -2e+41], z, If[LessEqual[t$95$1, Infinity], 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 +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))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   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)) < -2.00000000000000001e41

   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. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 17.9%

      \[\leadsto \color{blue}{z} \]

   if -2.00000000000000001e41 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 15.9%

      \[\leadsto \color{blue}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 10^{+101}:\\ \;\;\;\;\left(t + z\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2.5 \cdot 10^{+302}:\\ \;\;\;\;\left(a + t\right) + \left(\mathsf{fma}\left(\log y, x, z\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 1e+101)
     (+ (+ t z) (fma i y (fma (log c) (- b 0.5) (* (log y) x))))
     (if (<= t_1 2.5e+302)
       (+ (+ a t) (+ (fma (log y) x z) (* (log c) (- b 0.5))))
       (fma y i (fma (log c) (- b 0.5) (+ a z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= 1e+101) {
		tmp = (t + z) + fma(i, y, fma(log(c), (b - 0.5), (log(y) * x)));
	} else if (t_1 <= 2.5e+302) {
		tmp = (a + t) + (fma(log(y), x, z) + (log(c) * (b - 0.5)));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + z)));
	}
	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 <= 1e+101)
		tmp = Float64(Float64(t + z) + fma(i, y, fma(log(c), Float64(b - 0.5), Float64(log(y) * x))));
	elseif (t_1 <= 2.5e+302)
		tmp = Float64(Float64(a + t) + Float64(fma(log(y), x, z) + Float64(log(c) * Float64(b - 0.5))));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+101], N[(N[(t + z), $MachinePrecision] + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5e+302], N[(N[(a + t), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.9999999999999998e100

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 9.9999999999999998e100 < (+.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)) < 2.5e302

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 2.5e302 < (+.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 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 90.0

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 93.1%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 88.3%

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

Alternative 4: 56.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+298)
     (fma y i z)
     (if (<= t_1 -1e+69)
       (+ (+ t z) (* (log c) b))
       (fma y i (fma (log c) (- b 0.5) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+298) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -1e+69) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+298)
		tmp = fma(y, i, z);
	elseif (t_1 <= -1e+69)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(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, -5e+298], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -1e+69], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000003e298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 72.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -5.0000000000000003e298 < (+.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)) < -1.0000000000000001e69

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-log.f64 53.1

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 53.1%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -1.0000000000000001e69 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 55.0%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 55.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 55.0

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

   7. Applied rewrites 55.0%

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

Alternative 5: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+298)
     (fma y i z)
     (if (<= t_1 -1e+69)
       (+ (+ t z) (* (log c) b))
       (fma y i (fma (log c) b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+298) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -1e+69) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, fma(log(c), b, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+298)
		tmp = fma(y, i, z);
	elseif (t_1 <= -1e+69)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, fma(log(c), b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+298], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -1e+69], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000003e298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 72.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -5.0000000000000003e298 < (+.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)) < -1.0000000000000001e69

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-log.f64 53.1

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 53.1%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -1.0000000000000001e69 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 55.0%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 55.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 51.8%

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

Alternative 6: 48.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, -0.5, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+298)
     (fma y i z)
     (if (<= t_1 -5e+60)
       (+ (+ t z) (* (log c) b))
       (fma y i (fma (log c) -0.5 a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+298) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -5e+60) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, fma(log(c), -0.5, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+298)
		tmp = fma(y, i, z);
	elseif (t_1 <= -5e+60)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, fma(log(c), -0.5, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(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, -5e+298], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -5e+60], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * -0.5 + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000003e298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 72.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -5.0000000000000003e298 < (+.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)) < -4.99999999999999975e60

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-log.f64 52.9

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.9%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -4.99999999999999975e60 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 55.0%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 55.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 40.2%

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

Alternative 7: 46.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -5e+298)
     (fma y i z)
     (if (<= t_1 -5e+60) (+ (+ t z) (* (log c) b)) (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -5e+298) {
		tmp = fma(y, i, z);
	} else if (t_1 <= -5e+60) {
		tmp = (t + z) + (log(c) * b);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -5e+298)
		tmp = fma(y, i, z);
	elseif (t_1 <= -5e+60)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+298], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$1, -5e+60], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.0000000000000003e298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 72.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -5.0000000000000003e298 < (+.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)) < -4.99999999999999975e60

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-log.f64 52.9

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   8. Applied rewrites 52.9%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -4.99999999999999975e60 < (+.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. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 37.0

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

      2. +-commutative 37.0

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

      3. *-commutative 37.0

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

      4. +-commutative 37.0

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

      5. associate-+l+ 37.0

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

      6. +-commutative 37.0

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

   7. Applied rewrites 37.0%

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

Alternative 8: 33.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY)) (* i y) (if (<= t_1 -5e+60) z (fma y i a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double 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 <= -5e+60) {
		tmp = z;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -5e+60)
		tmp = z;
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -5e+60], z, N[(y * i + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   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)) < -4.99999999999999975e60

   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. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 17.9%

      \[\leadsto \color{blue}{z} \]

   if -4.99999999999999975e60 < (+.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. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 37.0

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

      2. +-commutative 37.0

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

      3. *-commutative 37.0

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

      4. +-commutative 37.0

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

      5. associate-+l+ 37.0

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

      6. +-commutative 37.0

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

   7. Applied rewrites 37.0%

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

Alternative 9: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 400:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      400.0)
   (fma y i (fma (log c) (- b 0.5) z))
   (fma y i (fma (log c) b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= 400.0) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = fma(y, i, fma(log(c), b, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= 400.0)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = fma(y, i, fma(log(c), b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 400.0], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 400

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative 55.3

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

      2. *-commutative 55.3

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

      3. +-commutative 55.3

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

      4. associate-+l+ 55.3

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

   7. Applied rewrites 55.3%

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

   if 400 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 53.7%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 53.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 53.6%

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

Alternative 10: 37.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -2e+41)
   (fma y i z)
   (fma y i a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -2e+41) {
		tmp = fma(y, i, z);
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -2e+41)
		tmp = fma(y, i, z);
	else
		tmp = fma(y, i, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -2e+41], N[(y * i + z), $MachinePrecision], N[(y * i + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000001e41

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 38.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 38.9

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

   7. Applied rewrites 38.9%

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

   if -2.00000000000000001e41 < (+.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. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative 36.9

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

      2. +-commutative 36.9

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

      3. *-commutative 36.9

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

      4. +-commutative 36.9

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

      5. associate-+l+ 36.9

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

      6. +-commutative 36.9

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

   7. Applied rewrites 36.9%

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

Alternative 11: 15.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+41}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -2e+41)
   z
   a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -2e+41) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-2d+41)) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -2e+41) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -2e+41:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -2e+41)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -2e+41)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -2e+41], z, a]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000001e41

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Applied rewrites 15.7%

      \[\leadsto \color{blue}{z} \]

   if -2.00000000000000001e41 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 15.9%

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

Alternative 12: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + t\right) + \left(\mathsf{fma}\left(\log y, x, z\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ a t) (+ (fma (log y) x z) (* (log c) (- b 0.5))))))
   (if (<= x -4.2e+119)
     t_1
     (if (<= x 5.3e+166) (fma y i (fma (log c) (- b 0.5) (+ a z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + t) + (fma(log(y), x, z) + (log(c) * (b - 0.5)));
	double tmp;
	if (x <= -4.2e+119) {
		tmp = t_1;
	} else if (x <= 5.3e+166) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + t) + Float64(fma(log(y), x, z) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (x <= -4.2e+119)
		tmp = t_1;
	elseif (x <= 5.3e+166)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + t), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e+119], t$95$1, If[LessEqual[x, 5.3e+166], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999966e119 or 5.3e166 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -4.19999999999999966e119 < x < 5.3e166

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.0%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 78.3%

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

Alternative 13: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b, \log y \cdot x\right)\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (fma (log c) b (* (log y) x)))))
   (if (<= x -4.5e+244)
     t_1
     (if (<= x 1.5e+187) (fma y i (fma (log c) (- b 0.5) (+ a z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, fma(log(c), b, (log(y) * x)));
	double tmp;
	if (x <= -4.5e+244) {
		tmp = t_1;
	} else if (x <= 1.5e+187) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, fma(log(c), b, Float64(log(y) * x)))
	tmp = 0.0
	if (x <= -4.5e+244)
		tmp = t_1;
	elseif (x <= 1.5e+187)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[c], $MachinePrecision] * b + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+244], t$95$1, If[LessEqual[x, 1.5e+187], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000003e244 or 1.5e187 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 82.3%

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

   if -4.5000000000000003e244 < x < 1.5e187

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.0%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 75.2%

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

Alternative 14: 70.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+274}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 2.6e+274) (fma y i (fma (log c) (- b 0.5) (+ a z))) (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 2.6e+274) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (a + z)));
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 2.6e+274)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(a + z)));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 2.6e+274], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999998e274

   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. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 89.1

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 85.5%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 70.2%

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

   if 2.5999999999999998e274 < 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 76.6

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

   5. Applied rewrites 76.6%

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

Alternative 15: 16.0% accurate, 234.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
4. Step-by-step derivation

5. Applied rewrites 16.0%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(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)))