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

Percentage Accurate: 99.8% → 99.8%

Time: 7.4s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.8%

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.8%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.8%

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

   17. lift-+.f64 N/A

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

   18. lift-*.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(z, a\right) \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, \mathsf{max}\left(z, a\right) + \left(t + \mathsf{min}\left(z, a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(x, \log y, t\right) + \mathsf{max}\left(z, a\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (if (<= (fmin z a) -1.3e+72)
  (fma (- b 0.5) (log c) (fma i y (+ (fmax z a) (+ t (fmin z a)))))
  (fma y i (fma (log c) (- b 0.5) (+ (fma x (log y) t) (fmax z a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (fmin(z, a) <= -1.3e+72) {
		tmp = fma((b - 0.5), log(c), fma(i, y, (fmax(z, a) + (t + fmin(z, a)))));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (fma(x, log(y), t) + fmax(z, a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (fmin(z, a) <= -1.3e+72)
		tmp = fma(Float64(b - 0.5), log(c), fma(i, y, Float64(fmax(z, a) + Float64(t + fmin(z, a)))));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(fma(x, log(y), t) + fmax(z, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[Min[z, a], $MachinePrecision], -1.3e+72], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + N[(N[Max[z, a], $MachinePrecision] + N[(t + N[Min[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision] + N[Max[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e72

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8%

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.5%

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

   if -1.2999999999999999e72 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.1%

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

   4. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 86.1%

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

   6. Applied rewrites 86.1%

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

Alternative 3: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := a + \left(t + \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \log c \cdot -0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, a + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (+ a (+ t (fma i y (fma x (log y) (* (log c) -0.5)))))))
  (if (<= x -1.65e+199)
    t_1
    (if (<= x 1.55e+239)
      (fma (- b 0.5) (log c) (fma i y (+ a (+ t 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(i, y, fma(x, log(y), (log(c) * -0.5))));
	double tmp;
	if (x <= -1.65e+199) {
		tmp = t_1;
	} else if (x <= 1.55e+239) {
		tmp = fma((b - 0.5), log(c), fma(i, y, (a + (t + z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(t + fma(i, y, fma(x, log(y), Float64(log(c) * -0.5)))))
	tmp = 0.0
	if (x <= -1.65e+199)
		tmp = t_1;
	elseif (x <= 1.55e+239)
		tmp = fma(Float64(b - 0.5), log(c), fma(i, y, Float64(a + Float64(t + z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(t + N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+199], t$95$1, If[LessEqual[x, 1.55e+239], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e199 or 1.55e239 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 86.1%

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 70.6%

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

   if -1.6499999999999999e199 < x < 1.55e239

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8%

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.5%

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

Alternative 4: 85.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, a + \left(t + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x \cdot \log y}{i \cdot y}\right) \cdot \left(y \cdot i\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (if (<= x 1.55e+239)
  (fma (- b 0.5) (log c) (fma i y (+ a (+ t z))))
  (* (+ 1.0 (/ (* x (log y)) (* i y))) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 1.55e+239) {
		tmp = fma((b - 0.5), log(c), fma(i, y, (a + (t + z))));
	} else {
		tmp = (1.0 + ((x * log(y)) / (i * y))) * (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 1.55e+239)
		tmp = fma(Float64(b - 0.5), log(c), fma(i, y, Float64(a + Float64(t + z))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * log(y)) / Float64(i * y))) * Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 1.55e+239], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + N[(a + N[(t + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] / N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55e239

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8%

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.5%

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

   if 1.55e239 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right)} \cdot \left(y \cdot i\right) \]

      6. lower-unsound-/.f64 32.5%

         \[\leadsto \left(1 + \color{blue}{\frac{1 \cdot t}{y \cdot i}}\right) \cdot \left(y \cdot i\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 N/A

          \[\leadsto \left(1 + \frac{x \cdot \log y}{i \cdot y}\right) \cdot \left(y \cdot i\right) \]

       4. lower-*.f64 32.8%

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

   11. Applied rewrites 32.8%

       \[\leadsto \left(1 + \color{blue}{\frac{x \cdot \log y}{i \cdot y}}\right) \cdot \left(y \cdot i\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.8%

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.8%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.8%

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

   17. lift-+.f64 N/A

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

   18. lift-*.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 84.5%

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

Alternative 6: 83.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 84.5%

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower--.f64 68.9%

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

7. Applied rewrites 68.9%

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

Alternative 7: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := t + \mathsf{max}\left(z, a\right)\\ \mathbf{if}\;b - 0.5 \leq -4 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, t\_1\right)\right)\\ \mathbf{elif}\;b - 0.5 \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(y, i, \mathsf{min}\left(z, a\right)\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(z, a\right) + \left(t + \left(\mathsf{min}\left(z, a\right) + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (+ t (fmax z a))))
  (if (<= (- b 0.5) -4e+92)
    (fma y i (fma (log c) (- b 0.5) t_1))
    (if (<= (- b 0.5) 2e+120)
      (+ (fma -0.5 (log c) (fma y i (fmin z a))) t_1)
      (+ (fmax z a) (+ t (+ (fmin z a) (* (log c) (- b 0.5)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t + fmax(z, a);
	double tmp;
	if ((b - 0.5) <= -4e+92) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), t_1));
	} else if ((b - 0.5) <= 2e+120) {
		tmp = fma(-0.5, log(c), fma(y, i, fmin(z, a))) + t_1;
	} else {
		tmp = fmax(z, a) + (t + (fmin(z, a) + (log(c) * (b - 0.5))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t + fmax(z, a))
	tmp = 0.0
	if (Float64(b - 0.5) <= -4e+92)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), t_1));
	elseif (Float64(b - 0.5) <= 2e+120)
		tmp = Float64(fma(-0.5, log(c), fma(y, i, fmin(z, a))) + t_1);
	else
		tmp = Float64(fmax(z, a) + Float64(t + Float64(fmin(z, a) + Float64(log(c) * Float64(b - 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t + N[Max[z, a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b - 0.5), $MachinePrecision], -4e+92], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b - 0.5), $MachinePrecision], 2e+120], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(y * i + N[Min[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[Max[z, a], $MachinePrecision] + N[(t + N[(N[Min[z, a], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 b #s(literal 1/2 binary64)) < -4.0000000000000002e92

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.1%

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

   4. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 86.1%

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

   6. Applied rewrites 86.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 71.1%

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

   if -4.0000000000000002e92 < (-.f64 b #s(literal 1/2 binary64)) < 2e120

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 84.5%

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 68.9%

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

   9. Applied rewrites 68.9%

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

   if 2e120 < (-.f64 b #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 84.5%

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 61.6%

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

   7. Applied rewrites 61.6%

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

Alternative 8: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(z, a\right) \leq -2.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{max}\left(z, a\right) + \left(t + \left(\mathsf{min}\left(z, a\right) + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, t + \mathsf{max}\left(z, a\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (if (<= (fmin z a) -2.4e+154)
  (+ (fmax z a) (+ t (+ (fmin z a) (* (log c) (- b 0.5)))))
  (fma y i (fma (log c) (- b 0.5) (+ t (fmax z a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (fmin(z, a) <= -2.4e+154) {
		tmp = fmax(z, a) + (t + (fmin(z, a) + (log(c) * (b - 0.5))));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (t + fmax(z, a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (fmin(z, a) <= -2.4e+154)
		tmp = Float64(fmax(z, a) + Float64(t + Float64(fmin(z, a) + Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(t + fmax(z, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[Min[z, a], $MachinePrecision], -2.4e+154], N[(N[Max[z, a], $MachinePrecision] + N[(t + N[(N[Min[z, a], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(t + N[Max[z, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000001e154

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 84.5%

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 61.6%

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

   7. Applied rewrites 61.6%

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

   if -2.4000000000000001e154 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.1%

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

   4. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 86.1%

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

   6. Applied rewrites 86.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 71.1%

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

Alternative 9: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ \mathbf{if}\;y \leq 3.7 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right) + \left(\mathsf{max}\left(z, t\right) + \left(t\_1 + \log c \cdot \left(b - 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_1}{i \cdot y}\right) \cdot \left(y \cdot i\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (fmin (fmin z t) a)))
  (if (<= y 3.7e+92)
    (+
     (fmax (fmin z t) a)
     (+ (fmax z t) (+ t_1 (* (log c) (- b 0.5)))))
    (* (+ 1.0 (/ t_1 (* i y))) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double tmp;
	if (y <= 3.7e+92) {
		tmp = fmax(fmin(z, t), a) + (fmax(z, t) + (t_1 + (log(c) * (b - 0.5))));
	} else {
		tmp = (1.0 + (t_1 / (i * y))) * (y * i);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = fmin(fmin(z, t), a)
    if (y <= 3.7d+92) then
        tmp = fmax(fmin(z, t), a) + (fmax(z, t) + (t_1 + (log(c) * (b - 0.5d0))))
    else
        tmp = (1.0d0 + (t_1 / (i * y))) * (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double tmp;
	if (y <= 3.7e+92) {
		tmp = fmax(fmin(z, t), a) + (fmax(z, t) + (t_1 + (Math.log(c) * (b - 0.5))));
	} else {
		tmp = (1.0 + (t_1 / (i * y))) * (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = fmin(fmin(z, t), a)
	tmp = 0
	if y <= 3.7e+92:
		tmp = fmax(fmin(z, t), a) + (fmax(z, t) + (t_1 + (math.log(c) * (b - 0.5))))
	else:
		tmp = (1.0 + (t_1 / (i * y))) * (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	tmp = 0.0
	if (y <= 3.7e+92)
		tmp = Float64(fmax(fmin(z, t), a) + Float64(fmax(z, t) + Float64(t_1 + Float64(log(c) * Float64(b - 0.5)))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_1 / Float64(i * y))) * Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = min(min(z, t), a);
	tmp = 0.0;
	if (y <= 3.7e+92)
		tmp = max(min(z, t), a) + (max(z, t) + (t_1 + (log(c) * (b - 0.5))));
	else
		tmp = (1.0 + (t_1 / (i * y))) * (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, If[LessEqual[y, 3.7e+92], N[(N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision] + N[(N[Max[z, t], $MachinePrecision] + N[(t$95$1 + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$1 / N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e92

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 84.5%

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 61.6%

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

   7. Applied rewrites 61.6%

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

   if 3.7e92 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

         \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right)} \cdot \left(y \cdot i\right) \]

      6. lower-unsound-/.f64 32.5%

         \[\leadsto \left(1 + \color{blue}{\frac{1 \cdot t}{y \cdot i}}\right) \cdot \left(y \cdot i\right) \]

   8. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(1 + \color{blue}{\frac{z}{i \cdot y}}\right) \cdot \left(y \cdot i\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(1 + \frac{z}{\color{blue}{i \cdot y}}\right) \cdot \left(y \cdot i\right) \]

       2. lower-*.f64 31.8%

          \[\leadsto \left(1 + \frac{z}{i \cdot \color{blue}{y}}\right) \cdot \left(y \cdot i\right) \]

   11. Applied rewrites 31.8%

       \[\leadsto \left(1 + \color{blue}{\frac{z}{i \cdot y}}\right) \cdot \left(y \cdot i\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + \mathsf{min}\left(z, a\right)\right) + t\right) + \mathsf{max}\left(z, a\right)\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\left(1 + \frac{\mathsf{min}\left(z, a\right)}{t}\right) \cdot t + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\mathsf{max}\left(z, a\right)}{t}\right) \cdot t + y \cdot i\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (if (<=
     (+
      (+
       (+ (+ (+ (* x (log y)) (fmin z a)) t) (fmax z a))
       (* (- b 0.5) (log c)))
      (* y i))
     -4e+24)
  (+ (* (+ 1.0 (/ (fmin z a) t)) t) (* y i))
  (+ (* (+ 1.0 (/ (fmax z a) t)) t) (* y i))))

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)) + fmin(z, a)) + t) + fmax(z, a)) + ((b - 0.5) * log(c))) + (y * i)) <= -4e+24) {
		tmp = ((1.0 + (fmin(z, a) / t)) * t) + (y * i);
	} else {
		tmp = ((1.0 + (fmax(z, a) / t)) * t) + (y * i);
	}
	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)) + fmin(z, a)) + t) + fmax(z, a)) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-4d+24)) then
        tmp = ((1.0d0 + (fmin(z, a) / t)) * t) + (y * i)
    else
        tmp = ((1.0d0 + (fmax(z, a) / t)) * t) + (y * i)
    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)) + fmin(z, a)) + t) + fmax(z, a)) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -4e+24) {
		tmp = ((1.0 + (fmin(z, a) / t)) * t) + (y * i);
	} else {
		tmp = ((1.0 + (fmax(z, a) / t)) * t) + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + fmin(z, a)) + t) + fmax(z, a)) + ((b - 0.5) * math.log(c))) + (y * i)) <= -4e+24:
		tmp = ((1.0 + (fmin(z, a) / t)) * t) + (y * i)
	else:
		tmp = ((1.0 + (fmax(z, a) / t)) * t) + (y * i)
	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)) + fmin(z, a)) + t) + fmax(z, a)) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -4e+24)
		tmp = Float64(Float64(Float64(1.0 + Float64(fmin(z, a) / t)) * t) + Float64(y * i));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(fmax(z, a) / t)) * t) + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + min(z, a)) + t) + max(z, a)) + ((b - 0.5) * log(c))) + (y * i)) <= -4e+24)
		tmp = ((1.0 + (min(z, a) / t)) * t) + (y * i);
	else
		tmp = ((1.0 + (max(z, a) / t)) * t) + (y * i);
	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] + N[Min[z, a], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] + N[Max[z, a], $MachinePrecision]), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -4e+24], N[(N[(N[(1.0 + N[(N[Min[z, a], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Max[z, a], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -3.9999999999999999e24

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in z around inf

      \[\leadsto \left(1 + \color{blue}{\frac{z}{t}}\right) \cdot t + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 46.0%

         \[\leadsto \left(1 + \frac{z}{\color{blue}{t}}\right) \cdot t + y \cdot i \]

   6. Applied rewrites 46.0%

      \[\leadsto \left(1 + \color{blue}{\frac{z}{t}}\right) \cdot t + y \cdot i \]

   if -3.9999999999999999e24 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in a around inf

      \[\leadsto \left(1 + \color{blue}{\frac{a}{t}}\right) \cdot t + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 46.9%

         \[\leadsto \left(1 + \frac{a}{\color{blue}{t}}\right) \cdot t + y \cdot i \]

   6. Applied rewrites 46.9%

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

Alternative 11: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + t\_1\right) + \mathsf{max}\left(z, t\right)\right) + t\_2\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{\mathsf{max}\left(z, t\right)}, \mathsf{max}\left(z, t\right), y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_2}{\mathsf{max}\left(z, t\right)}\right) \cdot \mathsf{max}\left(z, t\right) + y \cdot i\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (fmin (fmin z t) a)) (t_2 (fmax (fmin z t) a)))
  (if (<=
       (+
        (+
         (+ (+ (+ (* x (log y)) t_1) (fmax z t)) t_2)
         (* (- b 0.5) (log c)))
        (* y i))
       -4e+24)
    (fma (/ t_1 (fmax z t)) (fmax z t) (* y i))
    (+ (* (+ 1.0 (/ t_2 (fmax z t))) (fmax z t)) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double tmp;
	if (((((((x * log(y)) + t_1) + fmax(z, t)) + t_2) + ((b - 0.5) * log(c))) + (y * i)) <= -4e+24) {
		tmp = fma((t_1 / fmax(z, t)), fmax(z, t), (y * i));
	} else {
		tmp = ((1.0 + (t_2 / fmax(z, t))) * fmax(z, t)) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + fmax(z, t)) + t_2) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -4e+24)
		tmp = fma(Float64(t_1 / fmax(z, t)), fmax(z, t), Float64(y * i));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 / fmax(z, t))) * fmax(z, t)) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[Max[z, t], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -4e+24], N[(N[(t$95$1 / N[Max[z, t], $MachinePrecision]), $MachinePrecision] * N[Max[z, t], $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(t$95$2 / N[Max[z, t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[z, t], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -3.9999999999999999e24

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.5%

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.0%

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

   11. Applied rewrites 31.0%

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

   if -3.9999999999999999e24 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in a around inf

      \[\leadsto \left(1 + \color{blue}{\frac{a}{t}}\right) \cdot t + y \cdot i \]
   5. Step-by-step derivation

      1. lower-/.f64 46.9%

         \[\leadsto \left(1 + \frac{a}{\color{blue}{t}}\right) \cdot t + y \cdot i \]

   6. Applied rewrites 46.9%

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

Alternative 12: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_2 := \mathsf{max}\left(\mathsf{min}\left(z, t\right), a\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(z, t\right), t\_2\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + t\_1\right) + t\_3\right) + t\_4\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{t\_3}, t\_3, y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_4}{t\_3}, t\_3, y \cdot i\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (fmin (fmin z t) a))
       (t_2 (fmax (fmin z t) a))
       (t_3 (fmin (fmax z t) t_2))
       (t_4 (fmax (fmax z t) t_2)))
  (if (<=
       (+
        (+
         (+ (+ (+ (* x (log y)) t_1) t_3) t_4)
         (* (- b 0.5) (log c)))
        (* y i))
       -4e+24)
    (fma (/ t_1 t_3) t_3 (* y i))
    (fma (/ t_4 t_3) t_3 (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fmin(fmin(z, t), a);
	double t_2 = fmax(fmin(z, t), a);
	double t_3 = fmin(fmax(z, t), t_2);
	double t_4 = fmax(fmax(z, t), t_2);
	double tmp;
	if (((((((x * log(y)) + t_1) + t_3) + t_4) + ((b - 0.5) * log(c))) + (y * i)) <= -4e+24) {
		tmp = fma((t_1 / t_3), t_3, (y * i));
	} else {
		tmp = fma((t_4 / t_3), t_3, (y * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fmin(fmin(z, t), a)
	t_2 = fmax(fmin(z, t), a)
	t_3 = fmin(fmax(z, t), t_2)
	t_4 = fmax(fmax(z, t), t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + t_1) + t_3) + t_4) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -4e+24)
		tmp = fma(Float64(t_1 / t_3), t_3, Float64(y * i));
	else
		tmp = fma(Float64(t_4 / t_3), t_3, Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Min[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Min[z, t], $MachinePrecision], a], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[z, t], $MachinePrecision], t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -4e+24], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -3.9999999999999999e24

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.5%

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 31.0%

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

   11. Applied rewrites 31.0%

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

   if -3.9999999999999999e24 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.5%

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 31.8%

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

   11. Applied rewrites 31.8%

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

Alternative 13: 45.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ t_3 := \mathsf{min}\left(t, \mathsf{max}\left(z, a\right)\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{max}\left(t, \mathsf{max}\left(z, a\right)\right)}{t\_3}, t\_3, y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (* b (log c)))
       (t_2 (* (- b 0.5) (log c)))
       (t_3 (fmin t (fmax z a))))
  (if (<= t_2 -4e+146)
    t_1
    (if (<= t_2 2e+183)
      (fma (/ (fmax t (fmax z a)) t_3) t_3 (* y i))
      t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double t_3 = fmin(t, fmax(z, a));
	double tmp;
	if (t_2 <= -4e+146) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = fma((fmax(t, fmax(z, a)) / t_3), t_3, (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	t_3 = fmin(t, fmax(z, a))
	tmp = 0.0
	if (t_2 <= -4e+146)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = fma(Float64(fmax(t, fmax(z, a)) / t_3), t_3, Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Min[t, N[Max[z, a], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -4e+146], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(N[(N[Max[t, N[Max[z, a], $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$3 + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -3.9999999999999997e146 or 1.9999999999999999e183 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8%

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 17.0%

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

   9. Applied rewrites 17.0%

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

   if -3.9999999999999997e146 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 1.9999999999999999e183

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.5%

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 31.8%

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

   11. Applied rewrites 31.8%

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

Alternative 14: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{min}\left(t, a\right), y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
  :precision binary64
  (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
  (if (<= t_2 -1e+123)
    t_1
    (if (<= t_2 2e+183) (fma 1.0 (fmin t a) (* y i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -1e+123) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = fma(1.0, fmin(t, a), (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -1e+123)
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = fma(1.0, fmin(t, a), Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+123], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(1.0 * N[Min[t, a], $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -9.9999999999999998e122 or 1.9999999999999999e183 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8%

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 17.0%

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

   9. Applied rewrites 17.0%

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

   if -9.9999999999999998e122 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 1.9999999999999999e183

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. sum-to-mult N/A

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

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

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

   3. Applied rewrites 74.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 39.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 39.5%

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

   8. Applied rewrites 39.5%

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

Alternative 15: 39.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. sum-to-mult N/A

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

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

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

3. Applied rewrites 74.3%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 39.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 39.5%

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

8. Applied rewrites 39.5%

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

Alternative 16: 24.6% accurate, 5.4× speedup

Math

\[1 \cdot \left(y \cdot i\right) \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 1.0 * (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 = 1.0d0 * (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 1.0 * (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return 1.0 * (y * i)

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 1.0 * (y * i);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. sum-to-mult N/A

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

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

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

3. Applied rewrites 74.3%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 39.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. sum-to-mult N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

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

      \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right)} \cdot \left(y \cdot i\right) \]

   6. lower-unsound-/.f64 32.5%

      \[\leadsto \left(1 + \color{blue}{\frac{1 \cdot t}{y \cdot i}}\right) \cdot \left(y \cdot i\right) \]

8. Applied rewrites 32.5%

   \[\leadsto \color{blue}{\left(1 + \frac{1 \cdot t}{y \cdot i}\right) \cdot \left(y \cdot i\right)} \]

9. Taylor expanded in y around inf

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

11. Applied rewrites 24.6%

    \[\leadsto \color{blue}{1} \cdot \left(y \cdot i\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))