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

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. *-commutative N/A

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(b, a - 0.5, y\right)\right)\\ \mathbf{if}\;z \leq -0.27:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z (fma b (- a 0.5) y))))
   (if (<= z -0.27) t_1 (if (<= z 3.1e+72) (+ (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 = fma((1.0 - log(t)), z, fma(b, (a - 0.5), y));
	double tmp;
	if (z <= -0.27) {
		tmp = t_1;
	} else if (z <= 3.1e+72) {
		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 = fma(Float64(1.0 - log(t)), z, fma(b, Float64(a - 0.5), y))
	tmp = 0.0
	if (z <= -0.27)
		tmp = t_1;
	elseif (z <= 3.1e+72)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.27], t$95$1, If[LessEqual[z, 3.1e+72], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.27000000000000002 or 3.09999999999999988e72 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -0.27000000000000002 < z < 3.09999999999999988e72

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 96.5

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

   4. Applied rewrites 96.5%

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

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, t\_1\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -1.35e+166)
     (fma (- a 0.5) b t_1)
     (if (<= z 5.8e+130) (+ (fma (- a 0.5) b y) x) (+ (+ t_1 y) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -1.35e+166) {
		tmp = fma((a - 0.5), b, t_1);
	} else if (z <= 5.8e+130) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = (t_1 + y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -1.35e+166)
		tmp = fma(Float64(a - 0.5), b, t_1);
	elseif (z <= 5.8e+130)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(t_1 + y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.35e+166], N[(N[(a - 0.5), $MachinePrecision] * b + t$95$1), $MachinePrecision], If[LessEqual[z, 5.8e+130], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(t$95$1 + y), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35000000000000006e166

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 31.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 31.7

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

   6. Applied rewrites 31.7%

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

   7. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if -1.35000000000000006e166 < z < 5.7999999999999998e130

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 5.7999999999999998e130 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 75.1

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

   7. Applied rewrites 75.1%

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

Alternative 4: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+67}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(1 - \log t\right) \cdot z + y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -5e+67)
     t_2
     (if (<= t_1 2e+150) (+ (+ (* (- 1.0 (log t)) z) y) x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -5e+67) {
		tmp = t_2;
	} else if (t_1 <= 2e+150) {
		tmp = (((1.0 - log(t)) * z) + y) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -5e+67)
		tmp = t_2;
	elseif (t_1 <= 2e+150)
		tmp = Float64(Float64(Float64(Float64(1.0 - log(t)) * z) + y) + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+67], t$95$2, If[LessEqual[t$95$1, 2e+150], N[(N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.99999999999999976e67 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. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if -4.99999999999999976e67 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999996e150

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 90.7

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

   7. Applied rewrites 90.7%

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

Alternative 5: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot z\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+172}:\\ \;\;\;\;\left(x + z\right) - t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log t) z)))
   (if (<= z -8.5e+172)
     (- (+ x z) t_1)
     (if (<= z 1.25e+171) (+ (fma (- a 0.5) b y) x) (- (+ y z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(t) * z;
	double tmp;
	if (z <= -8.5e+172) {
		tmp = (x + z) - t_1;
	} else if (z <= 1.25e+171) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = (y + z) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(t) * z)
	tmp = 0.0
	if (z <= -8.5e+172)
		tmp = Float64(Float64(x + z) - t_1);
	elseif (z <= 1.25e+171)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(y + z) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -8.5e+172], N[(N[(x + z), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[z, 1.25e+171], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y + z), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.50000000000000053e172

   1. Initial program 99.7%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 67.5%

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

   if -8.50000000000000053e172 < z < 1.2500000000000001e171

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.4

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

   4. Applied rewrites 90.4%

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

   if 1.2500000000000001e171 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 69.8%

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

Alternative 6: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + z\right) - \log t \cdot z\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ x z) (* (log t) z))))
   (if (<= z -8.5e+172)
     t_1
     (if (<= z 4.6e+173) (+ (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 = (x + z) - (log(t) * z);
	double tmp;
	if (z <= -8.5e+172) {
		tmp = t_1;
	} else if (z <= 4.6e+173) {
		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(Float64(x + z) - Float64(log(t) * z))
	tmp = 0.0
	if (z <= -8.5e+172)
		tmp = t_1;
	elseif (z <= 4.6e+173)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.5e+172], t$95$1, If[LessEqual[z, 4.6e+173], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.50000000000000053e172 or 4.5999999999999999e173 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 76.5

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

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 67.9%

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

   if -8.50000000000000053e172 < z < 4.5999999999999999e173

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.3

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

   4. Applied rewrites 90.3%

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

Alternative 7: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -3.2e+182)
     t_1
     (if (<= z 1.35e+201) (+ (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 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -3.2e+182) {
		tmp = t_1;
	} else if (z <= 1.35e+201) {
		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(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -3.2e+182)
		tmp = t_1;
	elseif (z <= 1.35e+201)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.2e+182], t$95$1, If[LessEqual[z, 1.35e+201], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999997e182 or 1.35e201 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 63.8

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

   4. Applied rewrites 63.8%

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

   if -3.1999999999999997e182 < z < 1.35e201

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 89.1

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

   4. Applied rewrites 89.1%

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

Alternative 8: 78.6% accurate, 2.3× speedup

Math

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

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 Float64(fma(Float64(a - 0.5), b, y) + x)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift--.f64 78.6

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

4. Applied rewrites 78.6%

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

Alternative 9: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -5e-155)
   (fma (- a 0.5) b x)
   (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))) <= -5e-155) {
		tmp = fma((a - 0.5), b, x);
	} 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))) <= -5e-155)
		tmp = fma(Float64(a - 0.5), b, x);
	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], -5e-155], N[(N[(a - 0.5), $MachinePrecision] * b + x), $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))) < -4.9999999999999999e-155

   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 inf

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

   4. Applied rewrites 56.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 56.8

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

   6. Applied rewrites 56.8%

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

   if -4.9999999999999999e-155 < (-.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. Taylor expanded in x around inf

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

   4. Applied rewrites 57.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.7

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

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 58.3

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

      2. +-commutative 58.3

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

      3. *-commutative 58.3

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

      4. associate--l+ 58.3

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

   9. Applied rewrites 58.3%

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

Alternative 10: 57.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 1e+42) (fma (- a 0.5) b x) (fma a b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 1e+42) {
		tmp = fma((a - 0.5), b, x);
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 1e+42)
		tmp = fma(Float64(a - 0.5), b, x);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 1e+42], N[(N[(a - 0.5), $MachinePrecision] * b + x), $MachinePrecision], N[(a * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 1.00000000000000004e42

   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 inf

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

   4. Applied rewrites 57.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.3

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

   6. Applied rewrites 57.3%

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

   if 1.00000000000000004e42 < (+.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 inf

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

   4. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.0

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 57.7

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

      2. +-commutative 57.7

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

      3. *-commutative 57.7

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

      4. associate--l+ 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 49.4%

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

Alternative 11: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;x + y \leq 10^{+42}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -1e+45)
   (fma a b x)
   (if (<= (+ x y) -1e-12)
     (fma -0.5 b x)
     (if (<= (+ x y) 1e+42) (* (- a 0.5) b) (fma a b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -1e+45) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= -1e-12) {
		tmp = fma(-0.5, b, x);
	} else if ((x + y) <= 1e+42) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -1e+45)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= -1e-12)
		tmp = fma(-0.5, b, x);
	elseif (Float64(x + y) <= 1e+42)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e+45], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], -1e-12], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+42], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x y) < -9.9999999999999993e44

   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 inf

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

   4. Applied rewrites 55.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 55.5

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 47.7%

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

   if -9.9999999999999993e44 < (+.f64 x y) < -9.9999999999999998e-13

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.3

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

   6. Applied rewrites 57.3%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 28.8%

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

   if -9.9999999999999998e-13 < (+.f64 x y) < 1.00000000000000004e42

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 54.5

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

   4. Applied rewrites 54.5%

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

   if 1.00000000000000004e42 < (+.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 inf

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

   4. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.0

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 57.7

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

      2. +-commutative 57.7

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

      3. *-commutative 57.7

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

      4. associate--l+ 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 49.4%

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

Alternative 12: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -5e+292)
     (fma a b x)
     (if (<= t_1 -5e-155)
       (fma -0.5 b x)
       (if (<= t_1 2e+306) (fma -0.5 b y) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -5e+292) {
		tmp = fma(a, b, x);
	} else if (t_1 <= -5e-155) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 2e+306) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -5e+292)
		tmp = fma(a, b, x);
	elseif (t_1 <= -5e-155)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 2e+306)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+292], N[(a * b + x), $MachinePrecision], If[LessEqual[t$95$1, -5e-155], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999996e292

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 79.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 79.1

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 74.1%

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

   if -4.9999999999999996e292 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999999e-155

   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 inf

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

   4. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.3

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 38.1%

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

   if -4.9999999999999999e-155 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.00000000000000003e306

   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 inf

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

   4. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.5

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 51.8

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

      2. +-commutative 51.8

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

      3. *-commutative 51.8

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

      4. associate--l+ 51.8

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

   9. Applied rewrites 51.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 38.1%

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

   if 2.00000000000000003e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 90.6

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

   4. Applied rewrites 90.6%

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

Alternative 13: 46.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -5e+292)
     (fma a b x)
     (if (<= t_1 5e-107) (fma -0.5 b x) (fma a b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -5e+292) {
		tmp = fma(a, b, x);
	} else if (t_1 <= 5e-107) {
		tmp = fma(-0.5, b, x);
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -5e+292)
		tmp = fma(a, b, x);
	elseif (t_1 <= 5e-107)
		tmp = fma(-0.5, b, x);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+292], N[(a * b + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-107], N[(-0.5 * b + x), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999996e292

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 79.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 79.1

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 74.1%

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

   if -4.9999999999999996e292 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 4.99999999999999971e-107

   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 inf

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

   4. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.3

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 38.1%

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

   if 4.99999999999999971e-107 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 57.4

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

   6. Applied rewrites 57.4%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 57.9

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

      2. +-commutative 57.9

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

      3. *-commutative 57.9

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

      4. associate--l+ 57.9

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

   9. Applied rewrites 57.9%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 46.3%

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

Alternative 14: 45.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -2e+304)
     (* b a)
     (if (<= t_1 -5e-155)
       (fma -0.5 b x)
       (if (<= t_1 2e+306) (fma -0.5 b y) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -2e+304) {
		tmp = b * a;
	} else if (t_1 <= -5e-155) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 2e+306) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -2e+304)
		tmp = Float64(b * a);
	elseif (t_1 <= -5e-155)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 2e+306)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e+304], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -5e-155], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(-0.5 * b + y), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.9999999999999999e304 or 2.00000000000000003e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if -1.9999999999999999e304 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999999e-155

   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 inf

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

   4. Applied rewrites 51.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.1

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 38.1%

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

   if -4.9999999999999999e-155 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.00000000000000003e306

   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 inf

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

   4. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.5

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in y around inf

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

      1. associate--l+ 51.8

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

      2. +-commutative 51.8

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

      3. *-commutative 51.8

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

      4. associate--l+ 51.8

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

   9. Applied rewrites 51.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 38.1%

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

Alternative 15: 45.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -2e+304)
     (* b a)
     (if (<= t_1 5e-107) (fma -0.5 b x) (if (<= t_1 2e+306) y (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -2e+304) {
		tmp = b * a;
	} else if (t_1 <= 5e-107) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 2e+306) {
		tmp = y;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -2e+304)
		tmp = Float64(b * a);
	elseif (t_1 <= 5e-107)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 2e+306)
		tmp = y;
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e+304], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 5e-107], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], y, N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.9999999999999999e304 or 2.00000000000000003e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if -1.9999999999999999e304 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 4.99999999999999971e-107

   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 inf

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

   4. Applied rewrites 51.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 51.2

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

   6. Applied rewrites 51.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 38.1%

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

   if 4.99999999999999971e-107 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 2.00000000000000003e306

   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 inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 25.0%

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

Alternative 16: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1.2 \cdot 10^{+195}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+221}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1.2e+195) (* b a) (if (<= t_1 2e+221) (+ y x) (* b a)))))

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 <= -1.2e+195) {
		tmp = b * a;
	} else if (t_1 <= 2e+221) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-1.2d+195)) then
        tmp = b * a
    else if (t_1 <= 2d+221) then
        tmp = y + x
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -1.2e+195) {
		tmp = b * a;
	} else if (t_1 <= 2e+221) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -1.2e+195:
		tmp = b * a
	elif t_1 <= 2e+221:
		tmp = y + x
	else:
		tmp = b * a
	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 <= -1.2e+195)
		tmp = Float64(b * a);
	elseif (t_1 <= 2e+221)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -1.2e+195)
		tmp = b * a;
	elseif (t_1 <= 2e+221)
		tmp = y + x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1.2e+195], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 2e+221], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.2000000000000001e195 or 2.0000000000000001e221 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.0

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

   4. Applied rewrites 66.0%

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

   if -1.2000000000000001e195 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.0000000000000001e221

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto y + x \]
   6. Step-by-step derivation

   7. Applied rewrites 56.2%

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

Alternative 17: 39.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -5e-155) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b)) <= -5e-155) {
		tmp = x;
	} else {
		tmp = y;
	}
	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) :: tmp
    if (((((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)) <= (-5d-155)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((x + y) + z) - (z * Math.log(t))) + ((a - 0.5) * b)) <= -5e-155) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)) <= -5e-155:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b)) <= -5e-155)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b)) <= -5e-155)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], -5e-155], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999999e-155

   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 inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 22.1%

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

   if -4.9999999999999999e-155 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 22.1%

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

Alternative 18: 22.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

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

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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. *-commutative N/A

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around inf

   \[\leadsto y + x \]
6. Step-by-step derivation

7. Applied rewrites 42.9%

   \[\leadsto y + x \]
8. Add Preprocessing

Alternative 19: 21.9% accurate, 26.1× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

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 x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 21.9%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(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)))