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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 13

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(b * N[(a - 0.5), $MachinePrecision] + z), $MachinePrecision] - N[(N[(N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision] - y), $MachinePrecision] - 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. 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. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. associate--r+ N/A

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

   16. lower--.f64 N/A

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

   17. lower--.f64 99.9

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \log t\\ t_2 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;x + y \leq -6 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(x + z\right) - t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + z\right) - t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (log t))) (t_2 (* (- a 0.5) b)))
   (if (<= (+ x y) -6e+23) (+ (- (+ x z) t_1) t_2) (+ (- (+ y z) t_1) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * log(t);
	double t_2 = (a - 0.5) * b;
	double tmp;
	if ((x + y) <= -6e+23) {
		tmp = ((x + z) - t_1) + t_2;
	} else {
		tmp = ((y + z) - t_1) + 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 = z * log(t)
    t_2 = (a - 0.5d0) * b
    if ((x + y) <= (-6d+23)) then
        tmp = ((x + z) - t_1) + t_2
    else
        tmp = ((y + z) - t_1) + 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 = z * Math.log(t);
	double t_2 = (a - 0.5) * b;
	double tmp;
	if ((x + y) <= -6e+23) {
		tmp = ((x + z) - t_1) + t_2;
	} else {
		tmp = ((y + z) - t_1) + t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z * math.log(t)
	t_2 = (a - 0.5) * b
	tmp = 0
	if (x + y) <= -6e+23:
		tmp = ((x + z) - t_1) + t_2
	else:
		tmp = ((y + z) - t_1) + t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * log(t))
	t_2 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (Float64(x + y) <= -6e+23)
		tmp = Float64(Float64(Float64(x + z) - t_1) + t_2);
	else
		tmp = Float64(Float64(Float64(y + z) - t_1) + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z * log(t);
	t_2 = (a - 0.5) * b;
	tmp = 0.0;
	if ((x + y) <= -6e+23)
		tmp = ((x + z) - t_1) + t_2;
	else
		tmp = ((y + z) - t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -6e+23], N[(N[(N[(x + z), $MachinePrecision] - t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(y + z), $MachinePrecision] - t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -6.0000000000000002e23

   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 y around 0

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

      1. lower-+.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if -6.0000000000000002e23 < (+.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 77.9%

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

Alternative 3: 91.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;\left(\left(x + z\right) - z \cdot \log t\right) + t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, z\right) - \left(\left(\log t \cdot z - y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -4e+144)
     (+ (- (+ x z) (* z (log t))) t_1)
     (if (<= t_1 2e+204)
       (- (fma b -0.5 z) (- (- (* (log t) z) y) x))
       (fma (- a 0.5) b (+ y x))))))

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 <= -4e+144) {
		tmp = ((x + z) - (z * log(t))) + t_1;
	} else if (t_1 <= 2e+204) {
		tmp = fma(b, -0.5, z) - (((log(t) * z) - y) - x);
	} else {
		tmp = fma((a - 0.5), b, (y + x));
	}
	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 <= -4e+144)
		tmp = Float64(Float64(Float64(x + z) - Float64(z * log(t))) + t_1);
	elseif (t_1 <= 2e+204)
		tmp = Float64(fma(b, -0.5, z) - Float64(Float64(Float64(log(t) * z) - y) - x));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+144], N[(N[(N[(x + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 2e+204], N[(N[(b * -0.5 + z), $MachinePrecision] - N[(N[(N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision] - y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.00000000000000009e144

   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 y around 0

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

      1. lower-+.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if -4.00000000000000009e144 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999998e204

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 75.9%

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

   if 1.99999999999999998e204 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

Alternative 4: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(b, -0.5, z\right) - \left(\left(\log t \cdot z - y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -4e+190)
     (fma (- a 0.5) b y)
     (if (<= t_1 2e+204)
       (- (fma b -0.5 z) (- (- (* (log t) z) y) x))
       (fma (- a 0.5) b (+ y x))))))

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 <= -4e+190) {
		tmp = fma((a - 0.5), b, y);
	} else if (t_1 <= 2e+204) {
		tmp = fma(b, -0.5, z) - (((log(t) * z) - y) - x);
	} else {
		tmp = fma((a - 0.5), b, (y + x));
	}
	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 <= -4e+190)
		tmp = fma(Float64(a - 0.5), b, y);
	elseif (t_1 <= 2e+204)
		tmp = Float64(fma(b, -0.5, z) - Float64(Float64(Float64(log(t) * z) - y) - x));
	else
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+190], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+204], N[(N[(b * -0.5 + z), $MachinePrecision] - N[(N[(N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision] - y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000003e190

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.1%

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

   if -4.0000000000000003e190 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999998e204

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 75.9%

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

   if 1.99999999999999998e204 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

Alternative 5: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\left(x + \left(y + z\right)\right) - z \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -1e+124)
     t_2
     (if (<= t_1 5e+30) (- (+ x (+ y z)) (* z (log t))) 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 <= -1e+124) {
		tmp = t_2;
	} else if (t_1 <= 5e+30) {
		tmp = (x + (y + z)) - (z * log(t));
	} 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 <= -1e+124)
		tmp = t_2;
	elseif (t_1 <= 5e+30)
		tmp = Float64(Float64(x + Float64(y + z)) - Float64(z * log(t)));
	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, -1e+124], t$95$2, If[LessEqual[t$95$1, 5e+30], N[(N[(x + N[(y + z), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -9.99999999999999948e123 or 4.9999999999999998e30 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

   if -9.99999999999999948e123 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4.9999999999999998e30

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in b around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-log.f64 63.5

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

   11. Applied rewrites 63.5%

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

Alternative 6: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(1 - \log t\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- 1.0 (log t)))))
   (if (<= z -1.9e+178)
     t_1
     (if (<= z 2.4e+234) (fma (- a 0.5) b (+ y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (1.0 - log(t));
	double tmp;
	if (z <= -1.9e+178) {
		tmp = t_1;
	} else if (z <= 2.4e+234) {
		tmp = fma((a - 0.5), b, (y + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(1.0 - log(t)))
	tmp = 0.0
	if (z <= -1.9e+178)
		tmp = t_1;
	elseif (z <= 2.4e+234)
		tmp = fma(Float64(a - 0.5), b, Float64(y + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+178], t$95$1, If[LessEqual[z, 2.4e+234], N[(N[(a - 0.5), $MachinePrecision] * b + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999999e178 or 2.40000000000000011e234 < 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 21.4

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

   4. Applied rewrites 21.4%

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

   if -1.89999999999999999e178 < z < 2.40000000000000011e234

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

Alternative 7: 78.5% 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. 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. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. associate--r+ N/A

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

   16. lower--.f64 N/A

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

   17. lower--.f64 99.9

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.9

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
5. 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 79.5

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

6. Applied rewrites 79.5%

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
7. 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. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 79.5

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

8. Applied rewrites 79.5%

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

Alternative 8: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -4 \cdot 10^{-109}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -4e-109)
   (+ 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))) <= -4e-109) {
		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(x + y) + z) - Float64(z * log(t))) <= -4e-109)
		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[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-109], 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 x y) z) (*.f64 z (log.f64 t))) < -4e-109

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 58.3

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

   9. Applied rewrites 58.3%

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

   if -4e-109 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.1%

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

Alternative 9: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+53}:\\ \;\;\;\;x + y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4e+144) t_2 (if (<= t_1 1e+53) (+ x (* y 1.0)) 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 <= -4e+144) {
		tmp = t_2;
	} else if (t_1 <= 1e+53) {
		tmp = x + (y * 1.0);
	} 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 <= -4e+144)
		tmp = t_2;
	elseif (t_1 <= 1e+53)
		tmp = Float64(x + Float64(y * 1.0));
	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, -4e+144], t$95$2, If[LessEqual[t$95$1, 1e+53], N[(x + N[(y * 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.00000000000000009e144 or 9.9999999999999999e52 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - 0.5\right)\right)} \]
   7. 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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.1%

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

   if -4.00000000000000009e144 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.9999999999999999e52

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 69.7

         \[\leadsto x + y \cdot \left(1 + \frac{b \cdot \left(a - 0.5\right)}{y}\right) \]

   9. Applied rewrites 69.7%

      \[\leadsto x + y \cdot \color{blue}{\left(1 + \frac{b \cdot \left(a - 0.5\right)}{y}\right)} \]

   10. Taylor expanded in y around inf

       \[\leadsto x + y \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 43.6%

       \[\leadsto x + y \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+167}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+150}:\\ \;\;\;\;x + y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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+167) t_2 (if (<= t_1 5e+150) (+ x (* y 1.0)) 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+167) {
		tmp = t_2;
	} else if (t_1 <= 5e+150) {
		tmp = x + (y * 1.0);
	} 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+167)) then
        tmp = t_2
    else if (t_1 <= 5d+150) then
        tmp = x + (y * 1.0d0)
    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+167) {
		tmp = t_2;
	} else if (t_1 <= 5e+150) {
		tmp = x + (y * 1.0);
	} 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+167:
		tmp = t_2
	elif t_1 <= 5e+150:
		tmp = x + (y * 1.0)
	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+167)
		tmp = t_2;
	elseif (t_1 <= 5e+150)
		tmp = Float64(x + Float64(y * 1.0));
	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+167)
		tmp = t_2;
	elseif (t_1 <= 5e+150)
		tmp = x + (y * 1.0);
	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+167], t$95$2, If[LessEqual[t$95$1, 5e+150], N[(x + N[(y * 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e167 or 5.00000000000000009e150 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.1

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

   9. Applied rewrites 58.1%

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

   10. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 37.4

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

   12. Applied rewrites 37.4%

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

   if -1e167 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.00000000000000009e150

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 69.7

         \[\leadsto x + y \cdot \left(1 + \frac{b \cdot \left(a - 0.5\right)}{y}\right) \]

   9. Applied rewrites 69.7%

      \[\leadsto x + y \cdot \color{blue}{\left(1 + \frac{b \cdot \left(a - 0.5\right)}{y}\right)} \]

   10. Taylor expanded in y around inf

       \[\leadsto x + y \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 43.6%

       \[\leadsto x + y \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4e+144) t_2 (if (<= t_1 2e+150) (+ x (* -0.5 b)) 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 <= -4e+144) {
		tmp = t_2;
	} else if (t_1 <= 2e+150) {
		tmp = x + (-0.5 * b);
	} 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 <= (-4d+144)) then
        tmp = t_2
    else if (t_1 <= 2d+150) then
        tmp = x + ((-0.5d0) * b)
    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 <= -4e+144) {
		tmp = t_2;
	} else if (t_1 <= 2e+150) {
		tmp = x + (-0.5 * b);
	} 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 <= -4e+144:
		tmp = t_2
	elif t_1 <= 2e+150:
		tmp = x + (-0.5 * b)
	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 <= -4e+144)
		tmp = t_2;
	elseif (t_1 <= 2e+150)
		tmp = Float64(x + Float64(-0.5 * b));
	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 <= -4e+144)
		tmp = t_2;
	elseif (t_1 <= 2e+150)
		tmp = x + (-0.5 * b);
	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, -4e+144], t$95$2, If[LessEqual[t$95$1, 2e+150], N[(x + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.00000000000000009e144 or 1.99999999999999996e150 < (*.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. 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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 58.1

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

   9. Applied rewrites 58.1%

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

   10. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 37.4

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

   12. Applied rewrites 37.4%

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

   if -4.00000000000000009e144 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999996e150

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   5. 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 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 55.8

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

   9. Applied rewrites 55.8%

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

   10. Taylor expanded in y around 0

       \[\leadsto x + \frac{-1}{2} \cdot b \]
   11. Step-by-step derivation

       1. lower-*.f64 34.8

          \[\leadsto x + -0.5 \cdot b \]

   12. Applied rewrites 34.8%

       \[\leadsto x + -0.5 \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 37.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(a - 0.5\right) \end{array} \]

FPCore

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

C

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

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 = b * (a - 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * N[(a - 0.5), $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. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. associate--r+ N/A

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

   16. lower--.f64 N/A

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

   17. lower--.f64 99.9

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.9

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
5. 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 79.5

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

6. Applied rewrites 79.5%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 58.1

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

9. Applied rewrites 58.1%

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

10. Taylor expanded in y around 0

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 37.4

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

12. Applied rewrites 37.4%

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

Alternative 13: 25.5% 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 25.5

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

4. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2025149 
(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)))