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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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)
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
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)
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
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8%

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lift-+.f64 N/A

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

   8. associate--r+ N/A

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

   9. sub-negate N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 99.8%

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := z \cdot \log t\\ \mathbf{if}\;\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right) \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\left(\left(\mathsf{min}\left(x, y\right) + z\right) - t\_1\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, z - \left(t\_1 - \mathsf{max}\left(x, y\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (log t))))
   (if (<= (+ (fmin x y) (fmax x y)) -5e-195)
     (+ (- (+ (fmin x y) z) t_1) (* (- a 0.5) b))
     (fma (- a 0.5) b (- z (- t_1 (fmax x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * log(t);
	double tmp;
	if ((fmin(x, y) + fmax(x, y)) <= -5e-195) {
		tmp = ((fmin(x, y) + z) - t_1) + ((a - 0.5) * b);
	} else {
		tmp = fma((a - 0.5), b, (z - (t_1 - fmax(x, y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * log(t))
	tmp = 0.0
	if (Float64(fmin(x, y) + fmax(x, y)) <= -5e-195)
		tmp = Float64(Float64(Float64(fmin(x, y) + z) - t_1) + Float64(Float64(a - 0.5) * b));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(z - Float64(t_1 - fmax(x, y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision], -5e-195], N[(N[(N[(N[Min[x, y], $MachinePrecision] + z), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(z - N[(t$95$1 - N[Max[x, y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -5.00000000000000009e-195

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 77.7%

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

   4. Applied rewrites 77.7%

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

   if -5.00000000000000009e-195 < (+.f64 x y)

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.4%

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

   6. Applied rewrites 79.4%

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

Alternative 3: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := z \cdot \log t\\ \mathbf{if}\;\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right) \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \left(\mathsf{min}\left(x, y\right) + z\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, z - \left(t\_1 - \mathsf{max}\left(x, y\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (log t))))
   (if (<= (+ (fmin x y) (fmax x y)) -1e+51)
     (fma -0.5 b (- (+ (fmin x y) z) t_1))
     (fma (- a 0.5) b (- z (- t_1 (fmax x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * log(t);
	double tmp;
	if ((fmin(x, y) + fmax(x, y)) <= -1e+51) {
		tmp = fma(-0.5, b, ((fmin(x, y) + z) - t_1));
	} else {
		tmp = fma((a - 0.5), b, (z - (t_1 - fmax(x, y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * log(t))
	tmp = 0.0
	if (Float64(fmin(x, y) + fmax(x, y)) <= -1e+51)
		tmp = fma(-0.5, b, Float64(Float64(fmin(x, y) + z) - t_1));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(z - Float64(t_1 - fmax(x, y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision], -1e+51], N[(-0.5 * b + N[(N[(N[Min[x, y], $MachinePrecision] + z), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(z - N[(t$95$1 - N[Max[x, y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1e51

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2}}, b, x + y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 54.2%

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

   10. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-log.f64 53.3%

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

   12. Applied rewrites 53.3%

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

   if -1e51 < (+.f64 x y)

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.4%

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

   6. Applied rewrites 79.4%

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

Alternative 4: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a - 0.5, b, z - z \cdot \log t\right)\\ \mathbf{if}\;z \leq -1.52 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- a 0.5) b (- z (* z (log t))))))
   (if (<= z -1.52e+206)
     t_1
     (if (<= z 4.7e+145) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a - 0.5), b, (z - (z * log(t))));
	double tmp;
	if (z <= -1.52e+206) {
		tmp = t_1;
	} else if (z <= 4.7e+145) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a - 0.5), b, Float64(z - Float64(z * log(t))))
	tmp = 0.0
	if (z <= -1.52e+206)
		tmp = t_1;
	elseif (z <= 4.7e+145)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b + N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.52e+206], t$95$1, If[LessEqual[z, 4.7e+145], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52e206 or 4.7000000000000002e145 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.4%

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

   6. Applied rewrites 79.4%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 57.6%

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

   9. Applied rewrites 57.6%

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

   if -1.52e206 < z < 4.7000000000000002e145

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

Alternative 5: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(-0.5, b, z - \left(z \cdot \log t - y\right)\right)\\ \mathbf{if}\;z \leq -1.52 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma -0.5 b (- z (- (* z (log t)) y)))))
   (if (<= z -1.52e+206)
     t_1
     (if (<= z 3.6e+161) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(-0.5, b, (z - ((z * log(t)) - y)));
	double tmp;
	if (z <= -1.52e+206) {
		tmp = t_1;
	} else if (z <= 3.6e+161) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(-0.5, b, Float64(z - Float64(Float64(z * log(t)) - y)))
	tmp = 0.0
	if (z <= -1.52e+206)
		tmp = t_1;
	elseif (z <= 3.6e+161)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(-0.5 * b + N[(z - N[(N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.52e+206], t$95$1, If[LessEqual[z, 3.6e+161], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52e206 or 3.59999999999999984e161 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.4%

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

   6. Applied rewrites 79.4%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 54.9%

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

   if -1.52e206 < z < 3.59999999999999984e161

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

Alternative 6: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- 1.0 (log t)))))
   (if (<= z -7.8e+239)
     t_1
     (if (<= z 1.42e+184) (fma (- a 0.5) b (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - log(t));
	double tmp;
	if (z <= -7.8e+239) {
		tmp = t_1;
	} else if (z <= 1.42e+184) {
		tmp = fma((a - 0.5), b, (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(1.0 - log(t)))
	tmp = 0.0
	if (z <= -7.8e+239)
		tmp = t_1;
	elseif (z <= 1.42e+184)
		tmp = fma(Float64(a - 0.5), b, Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e+239], t$95$1, If[LessEqual[z, 1.42e+184], N[(N[(a - 0.5), $MachinePrecision] * b + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999996e239 or 1.42000000000000002e184 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 22.3%

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

   4. Applied rewrites 22.3%

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

   if -7.7999999999999996e239 < z < 1.42000000000000002e184

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

Alternative 7: 78.7% accurate, 2.3× speedup

Math

\[\mathsf{fma}\left(a - 0.5, b, x + y\right) \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8%

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lift-+.f64 N/A

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

   8. associate--r+ N/A

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

   9. sub-negate N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 99.8%

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in z around 0

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

   1. lower-+.f64 78.7%

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

6. Applied rewrites 78.7%

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

Alternative 8: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+259}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -2e+259)
     (* a b)
     (if (<= t_1 5e+291) (fma -0.5 b (+ x y)) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+259) {
		tmp = a * b;
	} else if (t_1 <= 5e+291) {
		tmp = fma(-0.5, b, (x + y));
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -2e+259)
		tmp = Float64(a * b);
	elseif (t_1 <= 5e+291)
		tmp = fma(-0.5, b, Float64(x + y));
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+259], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 5e+291], N[(-0.5 * b + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2e259 or 5.0000000000000001e291 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 26.2%

         \[\leadsto a \cdot \color{blue}{b} \]

   4. Applied rewrites 26.2%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -2e259 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000001e291

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2}}, b, x + y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 54.2%

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

Alternative 9: 48.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+52}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, \mathsf{max}\left(x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4.5e+52)
   (* a b)
   (if (<= a 2e+27) (fma -0.5 b (fmax x y)) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.5e+52) {
		tmp = a * b;
	} else if (a <= 2e+27) {
		tmp = fma(-0.5, b, fmax(x, y));
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4.5e+52)
		tmp = Float64(a * b);
	elseif (a <= 2e+27)
		tmp = fma(-0.5, b, fmax(x, y));
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4.5e+52], N[(a * b), $MachinePrecision], If[LessEqual[a, 2e+27], N[(-0.5 * b + N[Max[x, y], $MachinePrecision]), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e52 or 2e27 < a

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 26.2%

         \[\leadsto a \cdot \color{blue}{b} \]

   4. Applied rewrites 26.2%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -4.5e52 < a < 2e27

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8%

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. sub-negate N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 99.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 78.7%

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2}}, b, x + y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 54.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(-0.5, b, y\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 34.4%

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

Alternative 10: 26.2% accurate, 6.6× speedup

Math

\[a \cdot b \]

FPCore

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

C

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

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)
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
    code = a * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * b), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 26.2%

      \[\leadsto a \cdot \color{blue}{b} \]

4. Applied rewrites 26.2%

   \[\leadsto \color{blue}{a \cdot b} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))