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

Percentage Accurate: 99.9% → 99.9%

Time: 13.4s

Alternatives: 18

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-log.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 41.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

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 -6e+306)
     (* b a)
     (if (<= t_1 -1e-131) x (if (<= t_1 4e+291) (fma -0.5 b y) (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 <= -6e+306) {
		tmp = b * a;
	} else if (t_1 <= -1e-131) {
		tmp = x;
	} else if (t_1 <= 4e+291) {
		tmp = fma(-0.5, b, y);
	} 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 <= -6e+306)
		tmp = Float64(b * a);
	elseif (t_1 <= -1e-131)
		tmp = x;
	elseif (t_1 <= 4e+291)
		tmp = fma(-0.5, b, y);
	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, -6e+306], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -1e-131], x, If[LessEqual[t$95$1, 4e+291], N[(-0.5 * b + y), $MachinePrecision], N[(a * b + y), $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)) < -6.00000000000000042e306

   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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -6.00000000000000042e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.9999999999999999e-132

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 21.4%

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

   if -9.9999999999999999e-132 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 3.9999999999999998e291

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 49.1%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 49.1

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

      7. *-commutative 49.1

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

   10. Applied rewrites 49.1%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 43.0%

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

   if 3.9999999999999998e291 < (+.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.2%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 82.2

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

      7. *-commutative 82.2

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

   10. Applied rewrites 82.2%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 82.2%

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

Alternative 3: 35.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

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 -6e+306)
     (* b a)
     (if (<= t_1 -1e-131) x (if (<= t_1 INFINITY) (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 <= -6e+306) {
		tmp = b * a;
	} else if (t_1 <= -1e-131) {
		tmp = x;
	} else if (t_1 <= ((double) INFINITY)) {
		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 <= -6e+306)
		tmp = Float64(b * a);
	elseif (t_1 <= -1e-131)
		tmp = x;
	elseif (t_1 <= Inf)
		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, -6e+306], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -1e-131], x, If[LessEqual[t$95$1, Infinity], 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)) < -6.00000000000000042e306 or +inf.0 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -6.00000000000000042e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.9999999999999999e-132

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 21.4%

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

   if -9.9999999999999999e-132 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < +inf.0

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.2%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 55.2

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

      7. *-commutative 55.2

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

   10. Applied rewrites 55.2%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 39.7%

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

Alternative 4: 29.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+306}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

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 -6e+306)
     (* b a)
     (if (<= t_1 -5e-145) x (if (<= t_1 INFINITY) 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 <= -6e+306) {
		tmp = b * a;
	} else if (t_1 <= -5e-145) {
		tmp = x;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * Math.log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -6e+306) {
		tmp = b * a;
	} else if (t_1 <= -5e-145) {
		tmp = x;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)
	tmp = 0
	if t_1 <= -6e+306:
		tmp = b * a
	elif t_1 <= -5e-145:
		tmp = x
	elif t_1 <= math.inf:
		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 <= -6e+306)
		tmp = Float64(b * a);
	elseif (t_1 <= -5e-145)
		tmp = x;
	elseif (t_1 <= Inf)
		tmp = y;
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	tmp = 0.0;
	if (t_1 <= -6e+306)
		tmp = b * a;
	elseif (t_1 <= -5e-145)
		tmp = x;
	elseif (t_1 <= Inf)
		tmp = y;
	else
		tmp = b * a;
	end
	tmp_2 = 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, -6e+306], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -5e-145], x, If[LessEqual[t$95$1, Infinity], 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)) < -6.00000000000000042e306 or +inf.0 < (+.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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -6.00000000000000042e306 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -4.9999999999999998e-145

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 21.3%

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

   if -4.9999999999999998e-145 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < +inf.0

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 21.5%

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

Alternative 5: 47.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

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 -1e-131)
     (+ x (* a b))
     (if (<= t_1 4e+291) (fma -0.5 b y) (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 <= -1e-131) {
		tmp = x + (a * b);
	} else if (t_1 <= 4e+291) {
		tmp = fma(-0.5, b, y);
	} 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 <= -1e-131)
		tmp = Float64(x + Float64(a * b));
	elseif (t_1 <= 4e+291)
		tmp = fma(-0.5, b, y);
	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, -1e-131], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+291], N[(-0.5 * b + y), $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)) < -9.9999999999999999e-132

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 47.6%

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

   if -9.9999999999999999e-132 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 3.9999999999999998e291

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 49.1%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 49.1

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

      7. *-commutative 49.1

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

   10. Applied rewrites 49.1%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 43.0%

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

   if 3.9999999999999998e291 < (+.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.2%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 82.2

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

      7. *-commutative 82.2

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

   10. Applied rewrites 82.2%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 82.2%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t\_2 + x\\ \mathbf{elif}\;t\_1 \leq 10^{+154}:\\ \;\;\;\;\left(\mathsf{fma}\left(1 - \log t, z, y\right) + x\right) + -0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma (- a 0.5) b y)))
   (if (<= t_1 -5e+96)
     (+ t_2 x)
     (if (<= t_1 1e+154) (+ (+ (fma (- 1.0 (log t)) z y) x) (* -0.5 b)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y);
	double tmp;
	if (t_1 <= -5e+96) {
		tmp = t_2 + x;
	} else if (t_1 <= 1e+154) {
		tmp = (fma((1.0 - log(t)), z, y) + x) + (-0.5 * b);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(Float64(a - 0.5), b, y)
	tmp = 0.0
	if (t_1 <= -5e+96)
		tmp = Float64(t_2 + x);
	elseif (t_1 <= 1e+154)
		tmp = Float64(Float64(fma(Float64(1.0 - log(t)), z, y) + x) + Float64(-0.5 * b));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+96], N[(t$95$2 + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+154], N[(N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision] + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000004e96

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 98.4

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

   5. Applied rewrites 98.4%

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

   if -5.0000000000000004e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000004e154

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 95.5

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

   8. Applied rewrites 95.5%

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

   if 1.00000000000000004e154 < (*.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 96.9%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 96.9

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

      7. *-commutative 96.9

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

   10. Applied rewrites 96.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot b \leq 10^{+154}:\\ \;\;\;\;\left(\mathsf{fma}\left(1 - \log t, z, y\right) + x\right) + -0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t\_2 + x\\ \mathbf{elif}\;t\_1 \leq 10^{+154}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, b, z\right) + y\right) + x\right) - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma (- a 0.5) b y)))
   (if (<= t_1 -5e+96)
     (+ t_2 x)
     (if (<= t_1 1e+154) (- (+ (+ (fma -0.5 b z) y) x) (* (log t) z)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y);
	double tmp;
	if (t_1 <= -5e+96) {
		tmp = t_2 + x;
	} else if (t_1 <= 1e+154) {
		tmp = ((fma(-0.5, b, z) + y) + x) - (log(t) * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = fma(Float64(a - 0.5), b, y)
	tmp = 0.0
	if (t_1 <= -5e+96)
		tmp = Float64(t_2 + x);
	elseif (t_1 <= 1e+154)
		tmp = Float64(Float64(Float64(fma(-0.5, b, z) + y) + x) - Float64(log(t) * z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+96], N[(t$95$2 + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+154], N[(N[(N[(N[(-0.5 * b + z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000004e96

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 98.4

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

   5. Applied rewrites 98.4%

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

   if -5.0000000000000004e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000004e154

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if 1.00000000000000004e154 < (*.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 96.9%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 96.9

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

      7. *-commutative 96.9

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

   10. Applied rewrites 96.9%

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

Alternative 8: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+96} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + \left(z - \log t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -5e+96) (not (<= t_1 5e+102)))
     (+ (fma (- a 0.5) b y) x)
     (+ (+ y x) (- z (* (log t) z))))))

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 <= -5e+96) || !(t_1 <= 5e+102)) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = (y + x) + (z - (log(t) * z));
	}
	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 <= -5e+96) || !(t_1 <= 5e+102))
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(y + x) + Float64(z - Float64(log(t) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+96], N[Not[LessEqual[t$95$1, 5e+102]], $MachinePrecision]], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000004e96 or 5e102 < (*.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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 94.6

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

   5. Applied rewrites 94.6%

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

   if -5.0000000000000004e96 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5e102

   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. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   4. 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 91.0

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

   5. Applied rewrites 91.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-log.f64 N/A

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

      15. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{+96} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 5 \cdot 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + \left(z - \log t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -1e-131)
   (+ x (* (- a 0.5) b))
   (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))) <= -1e-131) {
		tmp = x + ((a - 0.5) * b);
	} 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))) <= -1e-131)
		tmp = Float64(x + Float64(Float64(a - 0.5) * b));
	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], -1e-131], N[(x + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -9.9999999999999999e-132

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 50.6%

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

   if -9.9999999999999999e-132 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.0%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 60.0

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

      7. *-commutative 60.0

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

   10. Applied rewrites 60.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 22.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -5e-145) 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-145) {
		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-145)) 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-145) {
		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-145:
		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-145)
		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-145)
		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-145], 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)) < -4.9999999999999998e-145

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 18.3%

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

   if -4.9999999999999998e-145 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 21.5%

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

Alternative 11: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -4 \cdot 10^{+58}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -4e+58)
   (+ x (* a b))
   (fma (- a 0.5) b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + y) + z) - (z * log(t))) <= -4e+58) {
		tmp = x + (a * b);
	} else {
		tmp = fma((a - 0.5), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) <= -4e+58)
		tmp = Float64(x + Float64(a * b));
	else
		tmp = fma(Float64(a - 0.5), b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+58], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -3.99999999999999978e58

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 40.2%

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

   if -3.99999999999999978e58 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.2%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 61.2

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

      7. *-commutative 61.2

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

   10. Applied rewrites 61.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -4 \cdot 10^{+58}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+216} \lor \neg \left(z \leq 9 \cdot 10^{+171}\right):\\ \;\;\;\;x + \left(z - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3e+216) (not (<= z 9e+171)))
   (+ x (- z (* (log t) z)))
   (+ (fma (- a 0.5) b y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3e+216) || !(z <= 9e+171)) {
		tmp = x + (z - (log(t) * z));
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3e+216) || !(z <= 9e+171))
		tmp = Float64(x + Float64(z - Float64(log(t) * z)));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3e+216], N[Not[LessEqual[z, 9e+171]], $MachinePrecision]], N[(x + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999998e216 or 8.99999999999999937e171 < 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. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   4. 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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative 84.3

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

   8. Applied rewrites 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-log.f64 N/A

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

      11. lift-*.f64 84.3

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

   10. Applied rewrites 84.3%

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

   if -2.9999999999999998e216 < z < 8.99999999999999937e171

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+216} \lor \neg \left(z \leq 9 \cdot 10^{+171}\right):\\ \;\;\;\;x + \left(z - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+242} \lor \neg \left(z \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.05e+242) (not (<= z 2.7e+172)))
   (- z (* (log t) z))
   (+ (fma (- a 0.5) b y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.05e+242) || !(z <= 2.7e+172)) {
		tmp = z - (log(t) * z);
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.05e+242) || !(z <= 2.7e+172))
		tmp = Float64(z - Float64(log(t) * z));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.05e+242], N[Not[LessEqual[z, 2.7e+172]], $MachinePrecision]], N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0499999999999999e242 or 2.7e172 < 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. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   4. 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 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto z - \color{blue}{\log t} \cdot z \]
   7. Step-by-step derivation

      1. +-commutative 81.3

         \[\leadsto z - \log t \cdot z \]

   8. Applied rewrites 81.3%

      \[\leadsto z - \color{blue}{\log t} \cdot z \]

   if -2.0499999999999999e242 < z < 2.7e172

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+242} \lor \neg \left(z \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;z - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+242} \lor \neg \left(z \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.05e+242) (not (<= z 2.7e+172)))
   (* (- 1.0 (log t)) z)
   (+ (fma (- a 0.5) b y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.05e+242) || !(z <= 2.7e+172)) {
		tmp = (1.0 - log(t)) * z;
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.05e+242) || !(z <= 2.7e+172))
		tmp = Float64(Float64(1.0 - log(t)) * z);
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.05e+242], N[Not[LessEqual[z, 2.7e+172]], $MachinePrecision]], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0499999999999999e242 or 2.7e172 < 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. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right)} \]
   4. 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 81.2

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

   5. Applied rewrites 81.2%

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

   if -2.0499999999999999e242 < z < 2.7e172

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+242} \lor \neg \left(z \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2000:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2000.0)
   (+ x (* -0.5 b))
   (if (<= (+ x y) 2e-44) (* (- 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) <= -2000.0) {
		tmp = x + (-0.5 * b);
	} else if ((x + y) <= 2e-44) {
		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) <= -2000.0)
		tmp = Float64(x + Float64(-0.5 * b));
	elseif (Float64(x + y) <= 2e-44)
		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], -2000.0], N[(x + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-44], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2e3

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 35.7

         \[\leadsto x + -0.5 \cdot \color{blue}{b} \]

   8. Applied rewrites 35.7%

      \[\leadsto x + \color{blue}{-0.5 \cdot b} \]

   if -2e3 < (+.f64 x y) < 1.99999999999999991e-44

   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. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 55.8

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

   5. Applied rewrites 55.8%

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

   if 1.99999999999999991e-44 < (+.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 54.4%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 54.4

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

      7. *-commutative 54.4

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 46.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2000:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2e+118)
   x
   (if (<= (+ x y) 2e-44) (* (- 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) <= -2e+118) {
		tmp = x;
	} else if ((x + y) <= 2e-44) {
		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) <= -2e+118)
		tmp = x;
	elseif (Float64(x + y) <= 2e-44)
		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], -2e+118], x, If[LessEqual[N[(x + y), $MachinePrecision], 2e-44], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1.99999999999999993e118

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 29.9%

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

   if -1.99999999999999993e118 < (+.f64 x y) < 1.99999999999999991e-44

   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. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 53.9

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

   5. Applied rewrites 53.9%

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

   if 1.99999999999999991e-44 < (+.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. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 54.4%

      \[\leadsto y + \left(a - 0.5\right) \cdot b \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 54.4

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

      7. *-commutative 54.4

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 46.0%

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

Alternative 17: 79.1% accurate, 9.7× 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. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
4. 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 76.8

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

5. Applied rewrites 76.8%

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

Alternative 18: 21.9% accurate, 126.0× 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. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Applied rewrites 21.5%

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

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + 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) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + 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) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025043 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 1/2) b)))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))