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

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

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: 99.9% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y + \mathsf{fma}\left(b, a - 0.5, z \cdot \left(1 - \log t\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ x (+ y (fma b (- a 0.5) (* z (- 1.0 (log t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + (y + fma(b, (a - 0.5), (z * (1.0 - log(t)))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(y + fma(b, Float64(a - 0.5), Float64(z * Float64(1.0 - log(t))))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(y + N[(b * N[(a - 0.5), $MachinePrecision] + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-log.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{x + \left(y + \mathsf{fma}\left(b, a - 0.5, z \cdot \left(1 - \log t\right)\right)\right)} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. sub-negate-rev N/A

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

   8. sub-flip-reverse N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 99.9

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

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 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
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 99.9

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

   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 99.9

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

   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 99.9

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

3. Applied rewrites 99.9%

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

Alternative 4: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{-60}:\\ \;\;\;\;x + \mathsf{fma}\left(b, a - 0.5, z \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(t\_1, z, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))))
   (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -5e-60)
     (+ x (fma b (- a 0.5) (* z t_1)))
     (fma (- a 0.5) b (fma t_1 z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double tmp;
	if (((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b)) <= -5e-60) {
		tmp = x + fma(b, (a - 0.5), (z * t_1));
	} else {
		tmp = fma((a - 0.5), b, fma(t_1, z, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b)) <= -5e-60)
		tmp = Float64(x + fma(b, Float64(a - 0.5), Float64(z * t_1)));
	else
		tmp = fma(Float64(a - 0.5), b, fma(t_1, z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -5e-60], N[(x + N[(b * N[(a - 0.5), $MachinePrecision] + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(t$95$1 * z + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -5.0000000000000001e-60

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -5.0000000000000001e-60 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 79.0

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. add-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. lower--.f64 79.0

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

   6. Applied rewrites 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 79.0

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

   8. Applied rewrites 79.0%

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{\mathsf{fma}\left(1 - \log t, z, y\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, \mathsf{fma}\left(1 - \log t, z, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -1e+77)
   (fma (- a 0.5) b (+ y x))
   (fma (- a 0.5) b (fma (- 1.0 (log t)) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -1e+77) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = fma((a - 0.5), b, fma((1.0 - log(t)), z, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -1e+77)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = fma(Float64(a - 0.5), b, fma(Float64(1.0 - log(t)), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e+77], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.99999999999999983e76

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   if -9.99999999999999983e76 < (+.f64 x y)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 79.0

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. add-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. lower--.f64 79.0

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

   6. Applied rewrites 79.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f64 N/A

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

      14. lower-fma.f64 79.0

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

   8. Applied rewrites 79.0%

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, \color{blue}{\mathsf{fma}\left(1 - \log t, z, y\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma (- a 0.5) b (+ y x))))
   (if (<= t_1 -2e+78)
     t_2
     (if (<= t_1 2e+92) (+ (fma (- 1.0 (log t)) z y) x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, (y + x));
	double tmp;
	if (t_1 <= -2e+78) {
		tmp = t_2;
	} else if (t_1 <= 2e+92) {
		tmp = fma((1.0 - log(t)), z, y) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(Float64(a - 0.5), b, Float64(y + x))
	tmp = 0.0
	if (t_1 <= -2e+78)
		tmp = t_2;
	elseif (t_1 <= 2e+92)
		tmp = Float64(fma(Float64(1.0 - log(t)), z, y) + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+78], t$95$2, If[LessEqual[t$95$1, 2e+92], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000002e78 or 2.0000000000000001e92 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   if -2.00000000000000002e78 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.0000000000000001e92

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 63.8

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

   7. Applied rewrites 63.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 63.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 63.8

         \[\leadsto \mathsf{fma}\left(1 - \log t, z, y\right) + x \]

   9. Applied rewrites 63.8%

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

Alternative 7: 80.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 8e+169) (fma (- a 0.5) b (+ y x)) (+ x (* z (- 1.0 (log t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 8e+169) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = x + (z * (1.0 - log(t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 8e+169)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(x + Float64(z * Float64(1.0 - log(t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 8e+169], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.99999999999999947e169

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   if 7.99999999999999947e169 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 43.2

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

   10. Applied rewrites 43.2%

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

Alternative 8: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 9.5e+169) (fma (- a 0.5) b (+ y x)) (* z (- 1.0 (log t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 9.5e+169) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = z * (1.0 - log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 9.5e+169)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = Float64(z * Float64(1.0 - log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 9.5e+169], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.4999999999999995e169

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   if 9.4999999999999995e169 < z

   1. Initial program 99.9%

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

   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 23.1

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

   4. Applied rewrites 23.1%

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 78.2

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

4. Applied rewrites 78.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 78.2

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

6. Applied rewrites 78.2%

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

Alternative 10: 71.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -5e-60], N[(x + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -5.0000000000000001e-60

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 57.2

         \[\leadsto x + b \cdot \left(a - 0.5\right) \]

   7. Applied rewrites 57.2%

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

   if -5.0000000000000001e-60 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 57.8%

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

Alternative 11: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma (- a 0.5) b y)))
   (if (<= t_1 -1e+101) t_2 (if (<= t_1 5e+115) (fma -0.5 b (+ y x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y);
	double tmp;
	if (t_1 <= -1e+101) {
		tmp = t_2;
	} else if (t_1 <= 5e+115) {
		tmp = fma(-0.5, b, (y + x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(Float64(a - 0.5), b, y)
	tmp = 0.0
	if (t_1 <= -1e+101)
		tmp = t_2;
	elseif (t_1 <= 5e+115)
		tmp = fma(-0.5, b, Float64(y + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+101], t$95$2, If[LessEqual[t$95$1, 5e+115], N[(-0.5 * b + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -9.9999999999999998e100 or 5.00000000000000008e115 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 57.8%

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

   if -9.9999999999999998e100 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.00000000000000008e115

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 54.5%

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

Alternative 12: 57.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a - 0.5, b, y\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 78.2

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

4. Applied rewrites 78.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 78.2

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

6. Applied rewrites 78.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 57.8%

   \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) \]
10. Add Preprocessing

Alternative 13: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+96}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (* b (- a 0.5))))
   (if (<= t_1 -1e+65) t_2 (if (<= t_1 1e+96) (* 1.0 y) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+65) {
		tmp = t_2;
	} else if (t_1 <= 1e+96) {
		tmp = 1.0 * y;
	} else {
		tmp = t_2;
	}
	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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    t_2 = b * (a - 0.5d0)
    if (t_1 <= (-1d+65)) then
        tmp = t_2
    else if (t_1 <= 1d+96) then
        tmp = 1.0d0 * y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+65) {
		tmp = t_2;
	} else if (t_1 <= 1e+96) {
		tmp = 1.0 * y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	t_2 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -1e+65:
		tmp = t_2
	elif t_1 <= 1e+96:
		tmp = 1.0 * y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -1e+65)
		tmp = t_2;
	elseif (t_1 <= 1e+96)
		tmp = Float64(1.0 * y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	t_2 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -1e+65)
		tmp = t_2;
	elseif (t_1 <= 1e+96)
		tmp = 1.0 * y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+65], t$95$2, If[LessEqual[t$95$1, 1e+96], N[(1.0 * y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -9.9999999999999999e64 or 1.00000000000000005e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.2

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

   6. Applied rewrites 78.2%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a - \frac{1}{2}\right)} \]

      2. lower--.f64 37.2

         \[\leadsto b \cdot \left(a - \color{blue}{0.5}\right) \]

   9. Applied rewrites 37.2%

      \[\leadsto \color{blue}{b \cdot \left(a - 0.5\right)} \]

   if -9.9999999999999999e64 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000005e96

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. sum-to-mult N/A

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

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

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 22.4%

      \[\leadsto \color{blue}{1} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t\_1 \leq 10^{+96}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1e+84) (* a b) (if (<= t_1 1e+96) (* 1.0 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 <= -1e+84) {
		tmp = a * b;
	} else if (t_1 <= 1e+96) {
		tmp = 1.0 * y;
	} else {
		tmp = a * b;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-1d+84)) then
        tmp = a * b
    else if (t_1 <= 1d+96) then
        tmp = 1.0d0 * y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static 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 <= -1e+84) {
		tmp = a * b;
	} else if (t_1 <= 1e+96) {
		tmp = 1.0 * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -1e+84:
		tmp = a * b
	elif t_1 <= 1e+96:
		tmp = 1.0 * 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 <= -1e+84)
		tmp = Float64(a * b);
	elseif (t_1 <= 1e+96)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -1e+84)
		tmp = a * b;
	elseif (t_1 <= 1e+96)
		tmp = 1.0 * y;
	else
		tmp = a * b;
	end
	tmp_2 = 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, -1e+84], N[(a * b), $MachinePrecision], If[LessEqual[t$95$1, 1e+96], N[(1.0 * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.00000000000000006e84 or 1.00000000000000005e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 25.4

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

   4. Applied rewrites 25.4%

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

   if -1.00000000000000006e84 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000005e96

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. sum-to-mult N/A

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

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

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 22.4%

      \[\leadsto \color{blue}{1} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 25.4% accurate, 6.6× speedup

Math

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

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 99.9%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 25.4

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

4. Applied rewrites 25.4%

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

Reproduce

herbie shell --seed 2025155 
(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)))