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

Percentage Accurate: 99.8% → 99.9%

Time: 7.8s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

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

Alternative 2: 46.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+191}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-113}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 + a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b))))
   (if (<= t_1 -2e+191)
     (+ x (* b a))
     (if (<= t_1 -1e-113)
       (+ x (* -0.5 b))
       (if (<= t_1 4e+297) (fma -0.5 b y) (* (+ -0.5 a) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
	double tmp;
	if (t_1 <= -2e+191) {
		tmp = x + (b * a);
	} else if (t_1 <= -1e-113) {
		tmp = x + (-0.5 * b);
	} else if (t_1 <= 4e+297) {
		tmp = fma(-0.5, b, y);
	} else {
		tmp = (-0.5 + a) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b))
	tmp = 0.0
	if (t_1 <= -2e+191)
		tmp = Float64(x + Float64(b * a));
	elseif (t_1 <= -1e-113)
		tmp = Float64(x + Float64(-0.5 * b));
	elseif (t_1 <= 4e+297)
		tmp = fma(-0.5, b, y);
	else
		tmp = Float64(Float64(-0.5 + a) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+191], N[(x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-113], N[(x + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+297], N[(-0.5 * b + y), $MachinePrecision], N[(N[(-0.5 + a), $MachinePrecision] * b), $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)) < -2.00000000000000015e191

   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 57.6%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

   8. Applied rewrites 54.8%

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

   if -2.00000000000000015e191 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.99999999999999979e-114

   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 x around inf

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{-1}{2}} \cdot b \]

      2. lower-*.f64 42.1

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

   8. Applied rewrites 42.1%

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

   if -9.99999999999999979e-114 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < 4.0000000000000001e297

   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 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 36.8%

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

   if 4.0000000000000001e297 < (+.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 b around inf

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

      1. distribute-rgt-out-- N/A

         \[\leadsto a \cdot b - \color{blue}{\frac{1}{2} \cdot b} \]

      2. fp-cancel-sub-sign N/A

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

      3. metadata-eval N/A

         \[\leadsto a \cdot b + \frac{-1}{2} \cdot b \]

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 81.3

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

   5. Applied rewrites 81.3%

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

Alternative 3: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+191}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, b, y\right)\\ \end{array} \end{array} \]

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 -2e+191)
     (+ x (* b a))
     (if (<= t_1 -1e-113) (fma -0.5 b (+ x y)) (fma (+ -0.5 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 <= -2e+191) {
		tmp = x + (b * a);
	} else if (t_1 <= -1e-113) {
		tmp = fma(-0.5, b, (x + y));
	} else {
		tmp = fma((-0.5 + 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 <= -2e+191)
		tmp = Float64(x + Float64(b * a));
	elseif (t_1 <= -1e-113)
		tmp = fma(-0.5, b, Float64(x + y));
	else
		tmp = fma(Float64(-0.5 + 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, -2e+191], N[(x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-113], N[(-0.5 * b + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + a), $MachinePrecision] * 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)) < -2.00000000000000015e191

   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 57.6%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

   8. Applied rewrites 54.8%

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

   if -2.00000000000000015e191 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -9.99999999999999979e-114

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

      6. associate--r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 70.8

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

   10. Applied rewrites 70.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 63.4%

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

   if -9.99999999999999979e-114 < (+.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 x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+191}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{elif}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \log t\\ t_2 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, b, t\_1 \cdot z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, y\right) + \mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- 1.0 (log t))) (t_2 (* (- a 0.5) b)))
   (if (<= t_2 -4e+206)
     (fma (+ -0.5 a) b (* t_1 z))
     (if (<= t_2 1e+100)
       (+ (fma t_1 z y) (fma -0.5 b x))
       (- (+ z (+ y x)) (* (- 0.5 a) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 - log(t);
	double t_2 = (a - 0.5) * b;
	double tmp;
	if (t_2 <= -4e+206) {
		tmp = fma((-0.5 + a), b, (t_1 * z));
	} else if (t_2 <= 1e+100) {
		tmp = fma(t_1, z, y) + fma(-0.5, b, x);
	} else {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 - log(t))
	t_2 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_2 <= -4e+206)
		tmp = fma(Float64(-0.5 + a), b, Float64(t_1 * z));
	elseif (t_2 <= 1e+100)
		tmp = Float64(fma(t_1, z, y) + fma(-0.5, b, x));
	else
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+206], N[(N[(-0.5 + a), $MachinePrecision] * b + N[(t$95$1 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+100], N[(N[(t$95$1 * z + y), $MachinePrecision] + N[(-0.5 * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e206

   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 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.2%

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

   if -4.0000000000000002e206 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000002e100

   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. *-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. associate-+l+ N/A

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

      11. associate-+r+ N/A

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

   5. Applied rewrites 96.2%

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

   if 1.00000000000000002e100 < (*.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 90.7

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

   7. Applied rewrites 90.7%

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

Alternative 5: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-21} \lor \neg \left(t\_1 \leq 10^{+100}\right):\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e-21) (not (<= t_1 1e+100)))
     (- (+ z (+ y x)) (* (- 0.5 a) b))
     (fma (- 1.0 (log t)) z (+ y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -2e-21) || !(t_1 <= 1e+100)) {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	} else {
		tmp = fma((1.0 - log(t)), z, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -2e-21) || !(t_1 <= 1e+100))
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	else
		tmp = fma(Float64(1.0 - log(t)), z, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-21], N[Not[LessEqual[t$95$1, 1e+100]], $MachinePrecision]], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999982e-21 or 1.00000000000000002e100 < (*.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -1.99999999999999982e-21 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000002e100

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

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

      2. associate-+r+ N/A

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

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

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

      4. *-lft-identity N/A

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

      5. associate--l+ N/A

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

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

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

      7. distribute-lft-neg-out N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 95.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{-21} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 10^{+100}\right):\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-21} \lor \neg \left(t\_1 \leq 10^{+100}\right):\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e-21) (not (<= t_1 1e+100)))
     (- (+ z (+ y x)) (* (- 0.5 a) b))
     (+ (fma (- 1.0 (log t)) z y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -2e-21) || !(t_1 <= 1e+100)) {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	} else {
		tmp = fma((1.0 - log(t)), z, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -2e-21) || !(t_1 <= 1e+100))
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	else
		tmp = Float64(fma(Float64(1.0 - log(t)), z, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-21], N[Not[LessEqual[t$95$1, 1e+100]], $MachinePrecision]], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999982e-21 or 1.00000000000000002e100 < (*.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -1.99999999999999982e-21 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000002e100

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

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

      2. associate-+r+ N/A

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

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

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

      4. *-lft-identity N/A

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

      5. associate--l+ N/A

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

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

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

      7. distribute-lft-neg-out N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 95.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 95.0

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

   7. Applied rewrites 95.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{-21} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 10^{+100}\right):\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -1 \cdot 10^{-113}:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -1e-113)
   (+ x (* (- a 0.5) b))
   (fma (+ -0.5 a) 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-113) {
		tmp = x + ((a - 0.5) * b);
	} else {
		tmp = fma((-0.5 + a), 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-113)
		tmp = Float64(x + Float64(Float64(a - 0.5) * b));
	else
		tmp = fma(Float64(-0.5 + a), 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-113], N[(x + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + a), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -9.99999999999999979e-114

   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 x around inf

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

   5. Applied rewrites 54.1%

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

   if -9.99999999999999979e-114 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.8%

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

Alternative 8: 22.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -1e-113) 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)) <= -1e-113) {
		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)) <= (-1d-113)) 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)) <= -1e-113) {
		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)) <= -1e-113:
		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)) <= -1e-113)
		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)) <= -1e-113)
		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], -1e-113], 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)) < -9.99999999999999979e-114

   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 x around inf

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

   5. Applied rewrites 21.1%

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

   if -9.99999999999999979e-114 < (+.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 y around inf

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

   5. Applied rewrites 22.2%

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

Alternative 9: 83.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 2e-10)
   (fma (- 1.0 (log t)) z (fma (+ -0.5 a) b x))
   (- (+ z (+ y x)) (* (- 0.5 a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 2e-10) {
		tmp = fma((1.0 - log(t)), z, fma((-0.5 + a), b, x));
	} else {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 2e-10)
		tmp = fma(Float64(1.0 - log(t)), z, fma(Float64(-0.5 + a), b, x));
	else
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 2e-10], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(-0.5 + a), $MachinePrecision] * b + x), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 2.00000000000000007e-10

   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 0

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

   5. Applied rewrites 82.2%

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

   if 2.00000000000000007e-10 < (+.f64 x y)

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 100.0

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 88.2

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

   7. Applied rewrites 88.2%

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

Alternative 10: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+180} \lor \neg \left(z \leq 3.9 \cdot 10^{+168}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.05e+180) (not (<= z 3.9e+168)))
   (fma (- 1.0 (log t)) z x)
   (- (+ z (+ y x)) (* (- 0.5 a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.05e+180) || !(z <= 3.9e+168)) {
		tmp = fma((1.0 - log(t)), z, x);
	} else {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.05e+180) || !(z <= 3.9e+168))
		tmp = fma(Float64(1.0 - log(t)), z, x);
	else
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.05e+180], N[Not[LessEqual[z, 3.9e+168]], $MachinePrecision]], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e180 or 3.89999999999999999e168 < z

   1. Initial program 99.6%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. fp-cancel-sub-sign-inv N/A

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

      2. associate-+r+ N/A

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

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

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

      4. *-lft-identity N/A

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

      5. associate--l+ N/A

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

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

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

      7. distribute-lft-neg-out N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(1 - \log t, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

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

   if -1.05e180 < z < 3.89999999999999999e168

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 90.3

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

   7. Applied rewrites 90.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+180} \lor \neg \left(z \leq 3.9 \cdot 10^{+168}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+182} \lor \neg \left(z \leq 1.1 \cdot 10^{+229}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.1e+182) (not (<= z 1.1e+229)))
   (* (- 1.0 (log t)) z)
   (- (+ z (+ y x)) (* (- 0.5 a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.1e+182) || !(z <= 1.1e+229)) {
		tmp = (1.0 - log(t)) * z;
	} else {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-2.1d+182)) .or. (.not. (z <= 1.1d+229))) then
        tmp = (1.0d0 - log(t)) * z
    else
        tmp = (z + (y + x)) - ((0.5d0 - a) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.1e+182) || !(z <= 1.1e+229)) {
		tmp = (1.0 - Math.log(t)) * z;
	} else {
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.1e+182) or not (z <= 1.1e+229):
		tmp = (1.0 - math.log(t)) * z
	else:
		tmp = (z + (y + x)) - ((0.5 - a) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.1e+182) || !(z <= 1.1e+229))
		tmp = Float64(Float64(1.0 - log(t)) * z);
	else
		tmp = Float64(Float64(z + Float64(y + x)) - Float64(Float64(0.5 - a) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.1e+182) || ~((z <= 1.1e+229)))
		tmp = (1.0 - log(t)) * z;
	else
		tmp = (z + (y + x)) - ((0.5 - a) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.1e+182], N[Not[LessEqual[z, 1.1e+229]], $MachinePrecision]], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e182 or 1.10000000000000002e229 < z

   1. Initial program 99.6%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   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. lower-log.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -2.0999999999999999e182 < z < 1.10000000000000002e229

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-*.f64 N/A

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

      18. lower-neg.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+182} \lor \neg \left(z \leq 1.1 \cdot 10^{+229}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(y + x\right)\right) - \left(0.5 - a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+206} \lor \neg \left(t\_1 \leq 10^{+213}\right):\\ \;\;\;\;\left(-0.5 + a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -4e+206) (not (<= t_1 1e+213))) (* (+ -0.5 a) b) (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -4e+206) || !(t_1 <= 1e+213)) {
		tmp = (-0.5 + a) * b;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if ((t_1 <= (-4d+206)) .or. (.not. (t_1 <= 1d+213))) then
        tmp = ((-0.5d0) + a) * b
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -4e+206) || !(t_1 <= 1e+213)) {
		tmp = (-0.5 + a) * b;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if (t_1 <= -4e+206) or not (t_1 <= 1e+213):
		tmp = (-0.5 + a) * b
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -4e+206) || !(t_1 <= 1e+213))
		tmp = Float64(Float64(-0.5 + a) * b);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if ((t_1 <= -4e+206) || ~((t_1 <= 1e+213)))
		tmp = (-0.5 + a) * b;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+206], N[Not[LessEqual[t$95$1, 1e+213]], $MachinePrecision]], N[(N[(-0.5 + a), $MachinePrecision] * b), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e206 or 9.99999999999999984e212 < (*.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 b around inf

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

      1. distribute-rgt-out-- N/A

         \[\leadsto a \cdot b - \color{blue}{\frac{1}{2} \cdot b} \]

      2. fp-cancel-sub-sign N/A

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

      3. metadata-eval N/A

         \[\leadsto a \cdot b + \frac{-1}{2} \cdot b \]

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -4.0000000000000002e206 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.99999999999999984e212

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

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

      2. associate-+r+ N/A

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

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

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

      4. *-lft-identity N/A

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

      5. associate--l+ N/A

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

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

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

      7. distribute-lft-neg-out N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto y + x \]

      2. lower-+.f64 59.1

         \[\leadsto y + x \]

   8. Applied rewrites 59.1%

      \[\leadsto y + \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -4 \cdot 10^{+206} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 10^{+213}\right):\\ \;\;\;\;\left(-0.5 + a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+206} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+247}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -4e+206) (not (<= t_1 4e+247))) (* b a) (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -4e+206) || !(t_1 <= 4e+247)) {
		tmp = b * a;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if ((t_1 <= (-4d+206)) .or. (.not. (t_1 <= 4d+247))) then
        tmp = b * a
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((t_1 <= -4e+206) || !(t_1 <= 4e+247)) {
		tmp = b * a;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if (t_1 <= -4e+206) or not (t_1 <= 4e+247):
		tmp = b * a
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if ((t_1 <= -4e+206) || !(t_1 <= 4e+247))
		tmp = Float64(b * a);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if ((t_1 <= -4e+206) || ~((t_1 <= 4e+247)))
		tmp = b * a;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+206], N[Not[LessEqual[t$95$1, 4e+247]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.0000000000000002e206 or 3.99999999999999981e247 < (*.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 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 81.4

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

   5. Applied rewrites 81.4%

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

   if -4.0000000000000002e206 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 3.99999999999999981e247

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

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

      2. associate-+r+ N/A

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

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

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

      4. *-lft-identity N/A

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

      5. associate--l+ N/A

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

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

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

      7. distribute-lft-neg-out N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto y + x \]

      2. lower-+.f64 56.8

         \[\leadsto y + x \]

   8. Applied rewrites 56.8%

      \[\leadsto y + \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -4 \cdot 10^{+206} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 4 \cdot 10^{+247}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{+51}:\\ \;\;\;\;x + b \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -2e+51) (+ x (* b a)) (fma (+ -0.5 a) b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -2e+51) {
		tmp = x + (b * a);
	} else {
		tmp = fma((-0.5 + a), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -2e+51)
		tmp = Float64(x + Float64(b * a));
	else
		tmp = fma(Float64(-0.5 + a), b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e+51], N[(x + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + a), $MachinePrecision] * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 x around inf

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.1

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

   8. Applied rewrites 44.1%

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

   if -2e51 < (+.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 x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 56.2%

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

Alternative 15: 80.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(z + N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - a), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. distribute-lft-neg-out N/A

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

   17. lift-*.f64 N/A

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

   18. lower-neg.f64 99.9

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. distribute-lft-out-- N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

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

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-neg-in N/A

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

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

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

   12. distribute-rgt-out-- N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower--.f64 79.2

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

7. Applied rewrites 79.2%

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

Alternative 16: 79.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. 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. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 78.4

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

5. Applied rewrites 78.4%

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

Alternative 17: 42.6% accurate, 31.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. 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. fp-cancel-sub-sign-inv N/A

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

   2. associate-+r+ N/A

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

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

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

   4. *-lft-identity N/A

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

   5. associate--l+ N/A

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

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

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

   7. distribute-lft-neg-out N/A

      \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(z \cdot \log t\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto 1 \cdot \left(\left(x + y\right) + \left(z + \left(\mathsf{neg}\left(\log t \cdot z\right)\right)\right)\right) \]

   9. distribute-lft-neg-out N/A

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

   10. mul-1-neg N/A

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

   11. distribute-rgt1-in N/A

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

   12. +-commutative N/A

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

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

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

   14. metadata-eval N/A

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

   15. *-lft-identity N/A

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

   16. *-commutative N/A

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

   17. remove-double-neg N/A

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

5. Applied rewrites 64.2%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto y + x \]

   2. lower-+.f64 44.6

      \[\leadsto y + x \]

8. Applied rewrites 44.6%

   \[\leadsto y + \color{blue}{x} \]
9. Add Preprocessing

Alternative 18: 22.8% 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 24.0%

   \[\leadsto \color{blue}{x} \]
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 2025026 
(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)))