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

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

Alternatives: 14

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. +-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. cancel-sign-sub-inv 99.9%

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

   10. distribute-rgt1-in 99.9%

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

   11. *-commutative 99.9%

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

   12. fma-def 99.9%

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

   13. +-commutative 99.9%

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

   14. unsub-neg 99.9%

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

   15. fma-def 99.9%

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

   16. sub-neg 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+185} \lor \neg \left(t_1 \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;t_1 + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (or (<= t_1 -1e+185) (not (<= t_1 2e+55)))
     (+ t_1 (- (+ x z) (* z (log t))))
     (+ x (+ y (* z (- 1.0 (log t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -1e+185) || !(t_1 <= 2e+55)) {
		tmp = t_1 + ((x + z) - (z * log(t)));
	} else {
		tmp = x + (y + (z * (1.0 - log(t))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = b * (a - 0.5d0)
    if ((t_1 <= (-1d+185)) .or. (.not. (t_1 <= 2d+55))) then
        tmp = t_1 + ((x + z) - (z * log(t)))
    else
        tmp = x + (y + (z * (1.0d0 - log(t))))
    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 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -1e+185) || !(t_1 <= 2e+55)) {
		tmp = t_1 + ((x + z) - (z * Math.log(t)));
	} else {
		tmp = x + (y + (z * (1.0 - Math.log(t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if (t_1 <= -1e+185) or not (t_1 <= 2e+55):
		tmp = t_1 + ((x + z) - (z * math.log(t)))
	else:
		tmp = x + (y + (z * (1.0 - math.log(t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if ((t_1 <= -1e+185) || !(t_1 <= 2e+55))
		tmp = Float64(t_1 + Float64(Float64(x + z) - Float64(z * log(t))));
	else
		tmp = Float64(x + Float64(y + Float64(z * Float64(1.0 - log(t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if ((t_1 <= -1e+185) || ~((t_1 <= 2e+55)))
		tmp = t_1 + ((x + z) - (z * log(t)));
	else
		tmp = x + (y + (z * (1.0 - log(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+185], N[Not[LessEqual[t$95$1, 2e+55]], $MachinePrecision]], N[(t$95$1 + N[(N[(x + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -9.9999999999999998e184 or 2.00000000000000002e55 < (*.f64 (-.f64 a 1/2) b)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.5%

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

      1. *-commutative 85.5%

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

   4. Simplified 85.5%

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

   if -9.9999999999999998e184 < (*.f64 (-.f64 a 1/2) b) < 2.00000000000000002e55

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 94.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+185} \lor \neg \left(b \cdot \left(a - 0.5\right) \leq 2 \cdot 10^{+55}\right):\\ \;\;\;\;b \cdot \left(a - 0.5\right) + \left(\left(x + z\right) - z \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \end{array} \]

Alternative 3: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+200} \lor \neg \left(t_1 \leq 10^{-15}\right):\\ \;\;\;\;\left(z + \left(x + y\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (or (<= t_1 -5e+200) (not (<= t_1 1e-15)))
     (+ (+ z (+ x y)) t_1)
     (+ x (+ y (* z (- 1.0 (log t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -5e+200) || !(t_1 <= 1e-15)) {
		tmp = (z + (x + y)) + t_1;
	} else {
		tmp = x + (y + (z * (1.0 - log(t))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = b * (a - 0.5d0)
    if ((t_1 <= (-5d+200)) .or. (.not. (t_1 <= 1d-15))) then
        tmp = (z + (x + y)) + t_1
    else
        tmp = x + (y + (z * (1.0d0 - log(t))))
    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 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -5e+200) || !(t_1 <= 1e-15)) {
		tmp = (z + (x + y)) + t_1;
	} else {
		tmp = x + (y + (z * (1.0 - Math.log(t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if (t_1 <= -5e+200) or not (t_1 <= 1e-15):
		tmp = (z + (x + y)) + t_1
	else:
		tmp = x + (y + (z * (1.0 - math.log(t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if ((t_1 <= -5e+200) || !(t_1 <= 1e-15))
		tmp = Float64(Float64(z + Float64(x + y)) + t_1);
	else
		tmp = Float64(x + Float64(y + Float64(z * Float64(1.0 - log(t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if ((t_1 <= -5e+200) || ~((t_1 <= 1e-15)))
		tmp = (z + (x + y)) + t_1;
	else
		tmp = x + (y + (z * (1.0 - log(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+200], N[Not[LessEqual[t$95$1, 1e-15]], $MachinePrecision]], N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(x + N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -5.00000000000000019e200 or 1.0000000000000001e-15 < (*.f64 (-.f64 a 1/2) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.8%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\sqrt[3]{z \cdot \log t} \cdot \sqrt[3]{z \cdot \log t}\right) \cdot \sqrt[3]{z \cdot \log t}}\right) + \left(a - 0.5\right) \cdot b \]

      2. pow3 99.8%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   3. Applied egg-rr 99.8%

      \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   4. Taylor expanded in z around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-+r+ 88.6%

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

   6. Simplified 88.6%

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

   if -5.00000000000000019e200 < (*.f64 (-.f64 a 1/2) b) < 1.0000000000000001e-15

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 93.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+200} \lor \neg \left(b \cdot \left(a - 0.5\right) \leq 10^{-15}\right):\\ \;\;\;\;\left(z + \left(x + y\right)\right) + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \end{array} \]

Alternative 4: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;t_1 + \left(z - z \cdot \log t\right)\\ \mathbf{elif}\;t_1 \leq 10^{-15}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (<= t_1 -1e+185)
     (+ t_1 (- z (* z (log t))))
     (if (<= t_1 1e-15)
       (+ x (+ y (* z (- 1.0 (log t)))))
       (+ (+ z (+ x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = t_1 + (z - (z * log(t)));
	} else if (t_1 <= 1e-15) {
		tmp = x + (y + (z * (1.0 - log(t))));
	} else {
		tmp = (z + (x + y)) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = b * (a - 0.5d0)
    if (t_1 <= (-1d+185)) then
        tmp = t_1 + (z - (z * log(t)))
    else if (t_1 <= 1d-15) then
        tmp = x + (y + (z * (1.0d0 - log(t))))
    else
        tmp = (z + (x + y)) + t_1
    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 = b * (a - 0.5);
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = t_1 + (z - (z * Math.log(t)));
	} else if (t_1 <= 1e-15) {
		tmp = x + (y + (z * (1.0 - Math.log(t))));
	} else {
		tmp = (z + (x + y)) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if t_1 <= -1e+185:
		tmp = t_1 + (z - (z * math.log(t)))
	elif t_1 <= 1e-15:
		tmp = x + (y + (z * (1.0 - math.log(t))))
	else:
		tmp = (z + (x + y)) + t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = Float64(t_1 + Float64(z - Float64(z * log(t))));
	elseif (t_1 <= 1e-15)
		tmp = Float64(x + Float64(y + Float64(z * Float64(1.0 - log(t)))));
	else
		tmp = Float64(Float64(z + Float64(x + y)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -1e+185)
		tmp = t_1 + (z - (z * log(t)));
	elseif (t_1 <= 1e-15)
		tmp = x + (y + (z * (1.0 - log(t))));
	else
		tmp = (z + (x + y)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(t$95$1 + N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(x + N[(y + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -9.9999999999999998e184

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.9%

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

      1. *-commutative 94.9%

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

   4. Simplified 94.9%

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

   5. Taylor expanded in x around 0 91.8%

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

   if -9.9999999999999998e184 < (*.f64 (-.f64 a 1/2) b) < 1.0000000000000001e-15

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 94.3%

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

   if 1.0000000000000001e-15 < (*.f64 (-.f64 a 1/2) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.7%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\sqrt[3]{z \cdot \log t} \cdot \sqrt[3]{z \cdot \log t}\right) \cdot \sqrt[3]{z \cdot \log t}}\right) + \left(a - 0.5\right) \cdot b \]

      2. pow3 99.8%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   3. Applied egg-rr 99.8%

      \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   4. Taylor expanded in z around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-+r+ 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;b \cdot \left(a - 0.5\right) + \left(z - z \cdot \log t\right)\\ \mathbf{elif}\;b \cdot \left(a - 0.5\right) \leq 10^{-15}:\\ \;\;\;\;x + \left(y + z \cdot \left(1 - \log t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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 + (x + y)) - (z * log(t))) + (b * (a - 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a - 0.5), $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. Final simplification 99.9%

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

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+219} \lor \neg \left(z \leq 7.5 \cdot 10^{+142}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.2e+219) (not (<= z 7.5e+142)))
   (+ x (* z (- 1.0 (log t))))
   (+ (+ z (+ x y)) (* b (- a 0.5)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.2e+219) || !(z <= 7.5e+142)) {
		tmp = x + (z * (1.0 - log(t)));
	} else {
		tmp = (z + (x + y)) + (b * (a - 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 <= (-1.2d+219)) .or. (.not. (z <= 7.5d+142))) then
        tmp = x + (z * (1.0d0 - log(t)))
    else
        tmp = (z + (x + y)) + (b * (a - 0.5d0))
    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 <= -1.2e+219) || !(z <= 7.5e+142)) {
		tmp = x + (z * (1.0 - Math.log(t)));
	} else {
		tmp = (z + (x + y)) + (b * (a - 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.2e+219) or not (z <= 7.5e+142):
		tmp = x + (z * (1.0 - math.log(t)))
	else:
		tmp = (z + (x + y)) + (b * (a - 0.5))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.2e+219) || !(z <= 7.5e+142))
		tmp = Float64(x + Float64(z * Float64(1.0 - log(t))));
	else
		tmp = Float64(Float64(z + Float64(x + y)) + Float64(b * Float64(a - 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.2e+219) || ~((z <= 7.5e+142)))
		tmp = x + (z * (1.0 - log(t)));
	else
		tmp = (z + (x + y)) + (b * (a - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.2e+219], N[Not[LessEqual[z, 7.5e+142]], $MachinePrecision]], N[(x + N[(z * N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e219 or 7.5000000000000002e142 < z

   1. Initial program 99.5%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. associate-+r+ 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. *-commutative 99.5%

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

      9. cancel-sign-sub-inv 99.5%

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

      10. distribute-rgt1-in 99.7%

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

      11. *-commutative 99.7%

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

      12. fma-def 99.7%

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

      13. +-commutative 99.7%

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

      14. unsub-neg 99.7%

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

      15. fma-def 99.7%

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

      16. sub-neg 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 86.4%

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

   if -1.2e219 < z < 7.5000000000000002e142

   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. Step-by-step derivation

      1. add-cube-cbrt 99.8%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\sqrt[3]{z \cdot \log t} \cdot \sqrt[3]{z \cdot \log t}\right) \cdot \sqrt[3]{z \cdot \log t}}\right) + \left(a - 0.5\right) \cdot b \]

      2. pow3 99.8%

         \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   3. Applied egg-rr 99.8%

      \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

   4. Taylor expanded in z around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. associate-+r+ 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+219} \lor \neg \left(z \leq 7.5 \cdot 10^{+142}\right):\\ \;\;\;\;x + z \cdot \left(1 - \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + \left(x + y\right)\right) + b \cdot \left(a - 0.5\right)\\ \end{array} \]

Alternative 7: 71.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+200} \lor \neg \left(t_1 \leq 10^{+171}\right):\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + -0.5 \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (- a 0.5))))
   (if (or (<= t_1 -5e+200) (not (<= t_1 1e+171)))
     (+ x t_1)
     (+ (+ x y) (* -0.5 b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -5e+200) || !(t_1 <= 1e+171)) {
		tmp = x + t_1;
	} else {
		tmp = (x + y) + (-0.5 * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = b * (a - 0.5d0)
    if ((t_1 <= (-5d+200)) .or. (.not. (t_1 <= 1d+171))) then
        tmp = x + t_1
    else
        tmp = (x + y) + ((-0.5d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (a - 0.5);
	double tmp;
	if ((t_1 <= -5e+200) || !(t_1 <= 1e+171)) {
		tmp = x + t_1;
	} else {
		tmp = (x + y) + (-0.5 * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (a - 0.5)
	tmp = 0
	if (t_1 <= -5e+200) or not (t_1 <= 1e+171):
		tmp = x + t_1
	else:
		tmp = (x + y) + (-0.5 * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(a - 0.5))
	tmp = 0.0
	if ((t_1 <= -5e+200) || !(t_1 <= 1e+171))
		tmp = Float64(x + t_1);
	else
		tmp = Float64(Float64(x + y) + Float64(-0.5 * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (a - 0.5);
	tmp = 0.0;
	if ((t_1 <= -5e+200) || ~((t_1 <= 1e+171)))
		tmp = x + t_1;
	else
		tmp = (x + y) + (-0.5 * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+200], N[Not[LessEqual[t$95$1, 1e+171]], $MachinePrecision]], N[(x + t$95$1), $MachinePrecision], N[(N[(x + y), $MachinePrecision] + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a 1/2) b) < -5.00000000000000019e200 or 9.99999999999999954e170 < (*.f64 (-.f64 a 1/2) b)

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 82.0%

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

   if -5.00000000000000019e200 < (*.f64 (-.f64 a 1/2) b) < 9.99999999999999954e170

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 70.2%

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

   3. Taylor expanded in a around 0 67.0%

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

      1. *-commutative 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+200} \lor \neg \left(b \cdot \left(a - 0.5\right) \leq 10^{+171}\right):\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) + -0.5 \cdot b\\ \end{array} \]

Alternative 8: 79.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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 + (x + y)) + (b * (a - 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
2. Step-by-step derivation

   1. add-cube-cbrt 99.4%

      \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(\sqrt[3]{z \cdot \log t} \cdot \sqrt[3]{z \cdot \log t}\right) \cdot \sqrt[3]{z \cdot \log t}}\right) + \left(a - 0.5\right) \cdot b \]

   2. pow3 99.4%

      \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

3. Applied egg-rr 99.4%

   \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{{\left(\sqrt[3]{z \cdot \log t}\right)}^{3}}\right) + \left(a - 0.5\right) \cdot b \]

4. Taylor expanded in z around 0 76.1%

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

   1. +-commutative 76.1%

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

   2. associate-+r+ 76.1%

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

6. Simplified 76.1%

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

7. Final simplification 76.1%

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

Alternative 9: 62.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.15e+135) (+ x (* b (- a 0.5))) (+ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.15e+135) {
		tmp = x + (b * (a - 0.5));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 (y <= 1.15d+135) then
        tmp = x + (b * (a - 0.5d0))
    else
        tmp = x + 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 (y <= 1.15e+135) {
		tmp = x + (b * (a - 0.5));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.15e+135:
		tmp = x + (b * (a - 0.5))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.15e+135)
		tmp = Float64(x + Float64(b * Float64(a - 0.5)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.15e+135)
		tmp = x + (b * (a - 0.5));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.15e+135], N[(x + N[(b * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1500000000000001e135

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 52.5%

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

   if 1.1500000000000001e135 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 64.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;x + b \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10: 78.6% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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) + (b * (a - 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0 75.1%

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

3. Final simplification 75.1%

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

Alternative 11: 51.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+107}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+129}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6.5e+107) (* a b) (if (<= b 7e+129) (+ x y) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.5e+107) {
		tmp = a * b;
	} else if (b <= 7e+129) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 (b <= (-6.5d+107)) then
        tmp = a * b
    else if (b <= 7d+129) then
        tmp = x + y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.5e+107) {
		tmp = a * b;
	} else if (b <= 7e+129) {
		tmp = x + y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6.5e+107:
		tmp = a * b
	elif b <= 7e+129:
		tmp = x + y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.5e+107)
		tmp = Float64(a * b);
	elseif (b <= 7e+129)
		tmp = Float64(x + y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6.5e+107)
		tmp = a * b;
	elseif (b <= 7e+129)
		tmp = x + y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.5e+107], N[(a * b), $MachinePrecision], If[LessEqual[b, 7e+129], N[(x + y), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.5000000000000006e107 or 6.9999999999999997e129 < b

   1. Initial program 100.0%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. cancel-sign-sub-inv 100.0%

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

      10. distribute-rgt1-in 100.0%

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

      11. *-commutative 100.0%

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

      12. fma-def 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

      15. fma-def 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around inf 53.2%

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

   5. Taylor expanded in x around 0 47.1%

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

   if -6.5000000000000006e107 < b < 6.9999999999999997e129

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 59.6%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+107}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+129}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 12: 50.0% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9000:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 9000.0) (+ x (* a b)) (+ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 9000.0) {
		tmp = x + (a * b);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 (y <= 9000.0d0) then
        tmp = x + (a * b)
    else
        tmp = x + 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 (y <= 9000.0) {
		tmp = x + (a * b);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 9000.0:
		tmp = x + (a * b)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 9000.0)
		tmp = Float64(x + Float64(a * b));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 9000.0)
		tmp = x + (a * b);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 9000.0], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9e3

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 40.5%

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

   if 9e3 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 55.3%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9000:\\ \;\;\;\;x + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 13: 30.5% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x -5.5e-22) x (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.5e-22) {
		tmp = x;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 <= (-5.5d-22)) then
        tmp = x
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.5e-22) {
		tmp = x;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -5.5e-22:
		tmp = x
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -5.5e-22)
		tmp = x;
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -5.5e-22)
		tmp = x;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -5.5e-22], x, N[(a * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000001e-22

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 46.0%

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

   5. Taylor expanded in x around inf 31.5%

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

   if -5.5000000000000001e-22 < x

   1. Initial program 99.9%

      \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. cancel-sign-sub-inv 99.9%

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

      10. distribute-rgt1-in 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

      15. fma-def 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 35.2%

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

   5. Taylor expanded in x around 0 21.4%

      \[\leadsto \color{blue}{a \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 14: 21.7% accurate, 115.0× 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

real(8) function code(x, y, z, t, a, b)
    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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-+r+ 99.9%

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

   6. +-commutative 99.9%

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

   7. +-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. cancel-sign-sub-inv 99.9%

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

   10. distribute-rgt1-in 99.9%

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

   11. *-commutative 99.9%

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

   12. fma-def 99.9%

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

   13. +-commutative 99.9%

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

   14. unsub-neg 99.9%

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

   15. fma-def 99.9%

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

   16. sub-neg 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in a around inf 38.3%

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

5. Taylor expanded in x around inf 20.6%

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

6. Final simplification 20.6%

   \[\leadsto x \]

Developer target: 99.6% 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

real(8) function code(x, y, z, t, a, b)
    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 2023224 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

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