Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 96.5%

Time: 10.0s

Alternatives: 13

Speedup: 0.8×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * 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)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * 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)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* z a) b))))
   (if (<= t_1 INFINITY) t_1 (* z (+ y (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((z * a) * b);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = z * (y + (a * b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((z * a) * b);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = z * (y + (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((x + (y * z)) + (t * a)) + ((z * a) * b)
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = z * (y + (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(z * a) * b))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(y + Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((x + (y * z)) + (t * a)) + ((z * a) * b);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = z * (y + (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < +inf.0

   1. Initial program 98.4%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l* 8.3%

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

      6. distribute-rgt-out 50.0%

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

      7. fma-def 58.3%

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

      8. +-commutative 58.3%

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

      9. remove-double-neg 58.3%

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

      10. sub-neg 58.3%

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

      11. *-commutative 58.3%

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

      12. fma-neg 58.3%

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

      13. remove-double-neg 58.3%

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

      14. +-commutative 58.3%

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

      15. fma-def 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in z around inf 91.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b \leq \infty:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(z \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 2: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+163} \lor \neg \left(a \leq 3.5 \cdot 10^{+86}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + z \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -3.8e+163) (not (<= a 3.5e+86)))
   (+ x (* a (+ t (* z b))))
   (+ (+ (+ x (* y z)) (* t a)) (* z (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -3.8e+163) || !(a <= 3.5e+86)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = ((x + (y * z)) + (t * a)) + (z * (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 ((a <= (-3.8d+163)) .or. (.not. (a <= 3.5d+86))) then
        tmp = x + (a * (t + (z * b)))
    else
        tmp = ((x + (y * z)) + (t * a)) + (z * (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 ((a <= -3.8e+163) || !(a <= 3.5e+86)) {
		tmp = x + (a * (t + (z * b)));
	} else {
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -3.8e+163) or not (a <= 3.5e+86):
		tmp = x + (a * (t + (z * b)))
	else:
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -3.8e+163) || !(a <= 3.5e+86))
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	else
		tmp = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(z * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -3.8e+163) || ~((a <= 3.5e+86)))
		tmp = x + (a * (t + (z * b)));
	else
		tmp = ((x + (y * z)) + (t * a)) + (z * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -3.8e+163], N[Not[LessEqual[a, 3.5e+86]], $MachinePrecision]], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.80000000000000008e163 or 3.50000000000000019e86 < a

   1. Initial program 85.4%

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

      1. associate-+l+ 85.4%

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

      2. +-commutative 85.4%

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

      3. *-commutative 85.4%

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

      4. *-commutative 85.4%

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

      5. associate-*l* 90.9%

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

      6. distribute-rgt-out 98.4%

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

      7. fma-def 98.4%

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

      8. +-commutative 98.4%

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

      9. remove-double-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. fma-neg 98.4%

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

      13. remove-double-neg 98.4%

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

      14. +-commutative 98.4%

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

      15. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 99.9%

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

   if -3.80000000000000008e163 < a < 3.50000000000000019e86

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 96.8%

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

   3. Simplified 96.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+163} \lor \neg \left(a \leq 3.5 \cdot 10^{+86}\right):\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + z \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 3: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + t \cdot a\\ t_2 := z \cdot \left(y + a \cdot b\right)\\ \mathbf{if}\;z \leq -170000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-33}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* t a))) (t_2 (* z (+ y (* a b)))))
   (if (<= z -170000.0)
     t_2
     (if (<= z -3.4e-12)
       t_1
       (if (<= z -2.5e-33)
         (* (* z a) b)
         (if (<= z -1.25e-148) (+ x (* y z)) (if (<= z 6.6e-61) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (t * a);
	double t_2 = z * (y + (a * b));
	double tmp;
	if (z <= -170000.0) {
		tmp = t_2;
	} else if (z <= -3.4e-12) {
		tmp = t_1;
	} else if (z <= -2.5e-33) {
		tmp = (z * a) * b;
	} else if (z <= -1.25e-148) {
		tmp = x + (y * z);
	} else if (z <= 6.6e-61) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x + (t * a)
    t_2 = z * (y + (a * b))
    if (z <= (-170000.0d0)) then
        tmp = t_2
    else if (z <= (-3.4d-12)) then
        tmp = t_1
    else if (z <= (-2.5d-33)) then
        tmp = (z * a) * b
    else if (z <= (-1.25d-148)) then
        tmp = x + (y * z)
    else if (z <= 6.6d-61) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (t * a);
	double t_2 = z * (y + (a * b));
	double tmp;
	if (z <= -170000.0) {
		tmp = t_2;
	} else if (z <= -3.4e-12) {
		tmp = t_1;
	} else if (z <= -2.5e-33) {
		tmp = (z * a) * b;
	} else if (z <= -1.25e-148) {
		tmp = x + (y * z);
	} else if (z <= 6.6e-61) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (t * a)
	t_2 = z * (y + (a * b))
	tmp = 0
	if z <= -170000.0:
		tmp = t_2
	elif z <= -3.4e-12:
		tmp = t_1
	elif z <= -2.5e-33:
		tmp = (z * a) * b
	elif z <= -1.25e-148:
		tmp = x + (y * z)
	elif z <= 6.6e-61:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(t * a))
	t_2 = Float64(z * Float64(y + Float64(a * b)))
	tmp = 0.0
	if (z <= -170000.0)
		tmp = t_2;
	elseif (z <= -3.4e-12)
		tmp = t_1;
	elseif (z <= -2.5e-33)
		tmp = Float64(Float64(z * a) * b);
	elseif (z <= -1.25e-148)
		tmp = Float64(x + Float64(y * z));
	elseif (z <= 6.6e-61)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (t * a);
	t_2 = z * (y + (a * b));
	tmp = 0.0;
	if (z <= -170000.0)
		tmp = t_2;
	elseif (z <= -3.4e-12)
		tmp = t_1;
	elseif (z <= -2.5e-33)
		tmp = (z * a) * b;
	elseif (z <= -1.25e-148)
		tmp = x + (y * z);
	elseif (z <= 6.6e-61)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -170000.0], t$95$2, If[LessEqual[z, -3.4e-12], t$95$1, If[LessEqual[z, -2.5e-33], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, -1.25e-148], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e-61], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.7e5 or 6.59999999999999992e-61 < z

   1. Initial program 89.7%

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

      1. associate-+l+ 89.7%

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

      2. +-commutative 89.7%

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

      3. *-commutative 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*l* 87.7%

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

      6. distribute-rgt-out 90.4%

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

      7. fma-def 91.1%

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

      8. +-commutative 91.1%

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

      9. remove-double-neg 91.1%

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

      10. sub-neg 91.1%

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

      11. *-commutative 91.1%

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

      12. fma-neg 91.1%

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

      13. remove-double-neg 91.1%

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

      14. +-commutative 91.1%

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

      15. fma-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in z around inf 80.0%

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

   if -1.7e5 < z < -3.4000000000000001e-12 or -1.25e-148 < z < 6.59999999999999992e-61

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 83.7%

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

   if -3.4000000000000001e-12 < z < -2.50000000000000014e-33

   1. Initial program 66.7%

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

      1. associate-+l+ 66.7%

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

      2. +-commutative 66.7%

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

      3. *-commutative 66.7%

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

      4. *-commutative 66.7%

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

      5. associate-*l* 66.1%

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

      6. distribute-rgt-out 99.5%

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

      7. fma-def 99.5%

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

      8. +-commutative 99.5%

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

      9. remove-double-neg 99.5%

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

      10. sub-neg 99.5%

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

      11. *-commutative 99.5%

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

      12. fma-neg 99.5%

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

      13. remove-double-neg 99.5%

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

      14. +-commutative 99.5%

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

      15. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in b around inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. add-log-exp 68.0%

         \[\leadsto \color{blue}{\log \left(e^{b \cdot \left(a \cdot z\right)}\right)} \]

      4. *-commutative 68.0%

         \[\leadsto \log \left(e^{\color{blue}{\left(a \cdot z\right) \cdot b}}\right) \]

      5. exp-prod 68.0%

         \[\leadsto \log \color{blue}{\left({\left(e^{a \cdot z}\right)}^{b}\right)} \]

      6. *-commutative 68.0%

         \[\leadsto \log \left({\left(e^{\color{blue}{z \cdot a}}\right)}^{b}\right) \]

      7. exp-prod 2.4%

         \[\leadsto \log \left({\color{blue}{\left({\left(e^{z}\right)}^{a}\right)}}^{b}\right) \]

   8. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\log \left({\left({\left(e^{z}\right)}^{a}\right)}^{b}\right)} \]
   9. Step-by-step derivation

      1. log-pow 2.4%

         \[\leadsto \color{blue}{b \cdot \log \left({\left(e^{z}\right)}^{a}\right)} \]

      2. log-pow 16.3%

         \[\leadsto b \cdot \color{blue}{\left(a \cdot \log \left(e^{z}\right)\right)} \]

      3. rem-log-exp 100.0%

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

   10. Simplified 100.0%

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

   if -2.50000000000000014e-33 < z < -1.25e-148

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 68.2%

      \[\leadsto \color{blue}{x + y \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -170000:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-33}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-61}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \end{array} \]

Alternative 4: 37.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+159}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* z b))))
   (if (<= z -5.2e-34)
     t_1
     (if (<= z 1.36e-291)
       x
       (if (<= z 1.66e-155)
         (* t a)
         (if (<= z 4.4e-61) x (if (<= z 3.5e+159) (* y z) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (z * b);
	double tmp;
	if (z <= -5.2e-34) {
		tmp = t_1;
	} else if (z <= 1.36e-291) {
		tmp = x;
	} else if (z <= 1.66e-155) {
		tmp = t * a;
	} else if (z <= 4.4e-61) {
		tmp = x;
	} else if (z <= 3.5e+159) {
		tmp = y * z;
	} else {
		tmp = 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 = a * (z * b)
    if (z <= (-5.2d-34)) then
        tmp = t_1
    else if (z <= 1.36d-291) then
        tmp = x
    else if (z <= 1.66d-155) then
        tmp = t * a
    else if (z <= 4.4d-61) then
        tmp = x
    else if (z <= 3.5d+159) then
        tmp = y * z
    else
        tmp = 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 = a * (z * b);
	double tmp;
	if (z <= -5.2e-34) {
		tmp = t_1;
	} else if (z <= 1.36e-291) {
		tmp = x;
	} else if (z <= 1.66e-155) {
		tmp = t * a;
	} else if (z <= 4.4e-61) {
		tmp = x;
	} else if (z <= 3.5e+159) {
		tmp = y * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (z * b)
	tmp = 0
	if z <= -5.2e-34:
		tmp = t_1
	elif z <= 1.36e-291:
		tmp = x
	elif z <= 1.66e-155:
		tmp = t * a
	elif z <= 4.4e-61:
		tmp = x
	elif z <= 3.5e+159:
		tmp = y * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(z * b))
	tmp = 0.0
	if (z <= -5.2e-34)
		tmp = t_1;
	elseif (z <= 1.36e-291)
		tmp = x;
	elseif (z <= 1.66e-155)
		tmp = Float64(t * a);
	elseif (z <= 4.4e-61)
		tmp = x;
	elseif (z <= 3.5e+159)
		tmp = Float64(y * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (z * b);
	tmp = 0.0;
	if (z <= -5.2e-34)
		tmp = t_1;
	elseif (z <= 1.36e-291)
		tmp = x;
	elseif (z <= 1.66e-155)
		tmp = t * a;
	elseif (z <= 4.4e-61)
		tmp = x;
	elseif (z <= 3.5e+159)
		tmp = y * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e-34], t$95$1, If[LessEqual[z, 1.36e-291], x, If[LessEqual[z, 1.66e-155], N[(t * a), $MachinePrecision], If[LessEqual[z, 4.4e-61], x, If[LessEqual[z, 3.5e+159], N[(y * z), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.1999999999999999e-34 or 3.4999999999999999e159 < z

   1. Initial program 83.2%

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

      1. associate-+l+ 83.2%

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

      2. +-commutative 83.2%

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

      3. *-commutative 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l* 84.1%

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

      6. distribute-rgt-out 89.5%

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

      7. fma-def 90.5%

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

      8. +-commutative 90.5%

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

      9. remove-double-neg 90.5%

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

      10. sub-neg 90.5%

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

      11. *-commutative 90.5%

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

      12. fma-neg 90.5%

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

      13. remove-double-neg 90.5%

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

      14. +-commutative 90.5%

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

      15. fma-def 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in b around inf 49.4%

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

      1. *-commutative 49.4%

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

   6. Simplified 49.4%

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

   if -5.1999999999999999e-34 < z < 1.36000000000000007e-291 or 1.65999999999999999e-155 < z < 4.40000000000000017e-61

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 52.9%

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

   if 1.36000000000000007e-291 < z < 1.65999999999999999e-155

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 59.7%

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

   if 4.40000000000000017e-61 < z < 3.4999999999999999e159

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 93.4%

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

      6. distribute-rgt-out 93.4%

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

      7. fma-def 93.4%

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

      8. +-commutative 93.4%

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

      9. remove-double-neg 93.4%

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

      10. sub-neg 93.4%

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

      11. *-commutative 93.4%

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

      12. fma-neg 93.4%

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

      13. remove-double-neg 93.4%

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

      14. +-commutative 93.4%

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

      15. fma-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around inf 48.0%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-34}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+159}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]

Alternative 5: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-34}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+162}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.6e-34)
   (* (* z a) b)
   (if (<= z 1.7e-291)
     x
     (if (<= z 8.8e-155)
       (* t a)
       (if (<= z 1.05e-60) x (if (<= z 1.4e+162) (* y z) (* a (* z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.6e-34) {
		tmp = (z * a) * b;
	} else if (z <= 1.7e-291) {
		tmp = x;
	} else if (z <= 8.8e-155) {
		tmp = t * a;
	} else if (z <= 1.05e-60) {
		tmp = x;
	} else if (z <= 1.4e+162) {
		tmp = y * z;
	} else {
		tmp = a * (z * 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 (z <= (-2.6d-34)) then
        tmp = (z * a) * b
    else if (z <= 1.7d-291) then
        tmp = x
    else if (z <= 8.8d-155) then
        tmp = t * a
    else if (z <= 1.05d-60) then
        tmp = x
    else if (z <= 1.4d+162) then
        tmp = y * z
    else
        tmp = a * (z * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.6e-34) {
		tmp = (z * a) * b;
	} else if (z <= 1.7e-291) {
		tmp = x;
	} else if (z <= 8.8e-155) {
		tmp = t * a;
	} else if (z <= 1.05e-60) {
		tmp = x;
	} else if (z <= 1.4e+162) {
		tmp = y * z;
	} else {
		tmp = a * (z * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.6e-34:
		tmp = (z * a) * b
	elif z <= 1.7e-291:
		tmp = x
	elif z <= 8.8e-155:
		tmp = t * a
	elif z <= 1.05e-60:
		tmp = x
	elif z <= 1.4e+162:
		tmp = y * z
	else:
		tmp = a * (z * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.6e-34)
		tmp = Float64(Float64(z * a) * b);
	elseif (z <= 1.7e-291)
		tmp = x;
	elseif (z <= 8.8e-155)
		tmp = Float64(t * a);
	elseif (z <= 1.05e-60)
		tmp = x;
	elseif (z <= 1.4e+162)
		tmp = Float64(y * z);
	else
		tmp = Float64(a * Float64(z * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.6e-34)
		tmp = (z * a) * b;
	elseif (z <= 1.7e-291)
		tmp = x;
	elseif (z <= 8.8e-155)
		tmp = t * a;
	elseif (z <= 1.05e-60)
		tmp = x;
	elseif (z <= 1.4e+162)
		tmp = y * z;
	else
		tmp = a * (z * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.6e-34], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 1.7e-291], x, If[LessEqual[z, 8.8e-155], N[(t * a), $MachinePrecision], If[LessEqual[z, 1.05e-60], x, If[LessEqual[z, 1.4e+162], N[(y * z), $MachinePrecision], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.5999999999999999e-34

   1. Initial program 86.0%

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

      1. associate-+l+ 86.0%

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

      2. +-commutative 86.0%

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

      3. *-commutative 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-*l* 86.1%

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

      6. distribute-rgt-out 89.2%

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

      7. fma-def 90.8%

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

      8. +-commutative 90.8%

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

      9. remove-double-neg 90.8%

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

      10. sub-neg 90.8%

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

      11. *-commutative 90.8%

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

      12. fma-neg 90.8%

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

      13. remove-double-neg 90.8%

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

      14. +-commutative 90.8%

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

      15. fma-def 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in b around inf 42.8%

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

      1. *-commutative 42.8%

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

   6. Simplified 42.8%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 45.9%

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

      2. *-commutative 45.9%

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

      3. add-log-exp 36.5%

         \[\leadsto \color{blue}{\log \left(e^{b \cdot \left(a \cdot z\right)}\right)} \]

      4. *-commutative 36.5%

         \[\leadsto \log \left(e^{\color{blue}{\left(a \cdot z\right) \cdot b}}\right) \]

      5. exp-prod 37.9%

         \[\leadsto \log \color{blue}{\left({\left(e^{a \cdot z}\right)}^{b}\right)} \]

      6. *-commutative 37.9%

         \[\leadsto \log \left({\left(e^{\color{blue}{z \cdot a}}\right)}^{b}\right) \]

      7. exp-prod 34.8%

         \[\leadsto \log \left({\color{blue}{\left({\left(e^{z}\right)}^{a}\right)}}^{b}\right) \]

   8. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{\log \left({\left({\left(e^{z}\right)}^{a}\right)}^{b}\right)} \]
   9. Step-by-step derivation

      1. log-pow 34.8%

         \[\leadsto \color{blue}{b \cdot \log \left({\left(e^{z}\right)}^{a}\right)} \]

      2. log-pow 35.5%

         \[\leadsto b \cdot \color{blue}{\left(a \cdot \log \left(e^{z}\right)\right)} \]

      3. rem-log-exp 45.9%

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

   10. Simplified 45.9%

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

   if -2.5999999999999999e-34 < z < 1.70000000000000013e-291 or 8.7999999999999996e-155 < z < 1.04999999999999996e-60

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 52.9%

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

   if 1.70000000000000013e-291 < z < 8.7999999999999996e-155

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 59.7%

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

   if 1.04999999999999996e-60 < z < 1.39999999999999995e162

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 93.4%

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

      6. distribute-rgt-out 93.4%

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

      7. fma-def 93.4%

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

      8. +-commutative 93.4%

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

      9. remove-double-neg 93.4%

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

      10. sub-neg 93.4%

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

      11. *-commutative 93.4%

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

      12. fma-neg 93.4%

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

      13. remove-double-neg 93.4%

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

      14. +-commutative 93.4%

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

      15. fma-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in y around inf 48.0%

      \[\leadsto \color{blue}{y \cdot z} \]

   if 1.39999999999999995e162 < z

   1. Initial program 77.1%

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

      1. associate-+l+ 77.1%

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

      2. +-commutative 77.1%

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

      3. *-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*l* 79.9%

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

      6. distribute-rgt-out 89.9%

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

      7. fma-def 89.9%

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

      8. +-commutative 89.9%

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

      9. remove-double-neg 89.9%

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

      10. sub-neg 89.9%

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

      11. *-commutative 89.9%

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

      12. fma-neg 89.9%

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

      13. remove-double-neg 89.9%

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

      14. +-commutative 89.9%

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

      15. fma-def 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in b around inf 63.5%

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

      1. *-commutative 63.5%

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

   6. Simplified 63.5%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-34}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+162}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]

Alternative 6: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot z\\ t_2 := \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;b \leq 4.65 \cdot 10^{+160}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* y z))) (t_2 (* (* z a) b)))
   (if (<= b -3.8e+64)
     t_2
     (if (<= b 9.5e-37)
       t_1
       (if (<= b 3.1e+102)
         (+ x (* t a))
         (if (<= b 4.65e+160) (* a (* z b)) (if (<= b 4.6e+213) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (y * z);
	double t_2 = (z * a) * b;
	double tmp;
	if (b <= -3.8e+64) {
		tmp = t_2;
	} else if (b <= 9.5e-37) {
		tmp = t_1;
	} else if (b <= 3.1e+102) {
		tmp = x + (t * a);
	} else if (b <= 4.65e+160) {
		tmp = a * (z * b);
	} else if (b <= 4.6e+213) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x + (y * z)
    t_2 = (z * a) * b
    if (b <= (-3.8d+64)) then
        tmp = t_2
    else if (b <= 9.5d-37) then
        tmp = t_1
    else if (b <= 3.1d+102) then
        tmp = x + (t * a)
    else if (b <= 4.65d+160) then
        tmp = a * (z * b)
    else if (b <= 4.6d+213) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (y * z);
	double t_2 = (z * a) * b;
	double tmp;
	if (b <= -3.8e+64) {
		tmp = t_2;
	} else if (b <= 9.5e-37) {
		tmp = t_1;
	} else if (b <= 3.1e+102) {
		tmp = x + (t * a);
	} else if (b <= 4.65e+160) {
		tmp = a * (z * b);
	} else if (b <= 4.6e+213) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (y * z)
	t_2 = (z * a) * b
	tmp = 0
	if b <= -3.8e+64:
		tmp = t_2
	elif b <= 9.5e-37:
		tmp = t_1
	elif b <= 3.1e+102:
		tmp = x + (t * a)
	elif b <= 4.65e+160:
		tmp = a * (z * b)
	elif b <= 4.6e+213:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(y * z))
	t_2 = Float64(Float64(z * a) * b)
	tmp = 0.0
	if (b <= -3.8e+64)
		tmp = t_2;
	elseif (b <= 9.5e-37)
		tmp = t_1;
	elseif (b <= 3.1e+102)
		tmp = Float64(x + Float64(t * a));
	elseif (b <= 4.65e+160)
		tmp = Float64(a * Float64(z * b));
	elseif (b <= 4.6e+213)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (y * z);
	t_2 = (z * a) * b;
	tmp = 0.0;
	if (b <= -3.8e+64)
		tmp = t_2;
	elseif (b <= 9.5e-37)
		tmp = t_1;
	elseif (b <= 3.1e+102)
		tmp = x + (t * a);
	elseif (b <= 4.65e+160)
		tmp = a * (z * b);
	elseif (b <= 4.6e+213)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -3.8e+64], t$95$2, If[LessEqual[b, 9.5e-37], t$95$1, If[LessEqual[b, 3.1e+102], N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.65e+160], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.6e+213], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.8000000000000001e64 or 4.59999999999999996e213 < b

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. +-commutative 87.4%

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

      3. *-commutative 87.4%

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

      4. *-commutative 87.4%

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

      5. associate-*l* 76.9%

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

      6. distribute-rgt-out 81.6%

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

      7. fma-def 81.6%

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

      8. +-commutative 81.6%

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

      9. remove-double-neg 81.6%

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

      10. sub-neg 81.6%

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

      11. *-commutative 81.6%

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

      12. fma-neg 81.6%

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

      13. remove-double-neg 81.6%

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

      14. +-commutative 81.6%

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

      15. fma-def 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in b around inf 66.4%

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

      1. *-commutative 66.4%

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

   6. Simplified 66.4%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. add-log-exp 41.7%

         \[\leadsto \color{blue}{\log \left(e^{b \cdot \left(a \cdot z\right)}\right)} \]

      4. *-commutative 41.7%

         \[\leadsto \log \left(e^{\color{blue}{\left(a \cdot z\right) \cdot b}}\right) \]

      5. exp-prod 39.7%

         \[\leadsto \log \color{blue}{\left({\left(e^{a \cdot z}\right)}^{b}\right)} \]

      6. *-commutative 39.7%

         \[\leadsto \log \left({\left(e^{\color{blue}{z \cdot a}}\right)}^{b}\right) \]

      7. exp-prod 37.7%

         \[\leadsto \log \left({\color{blue}{\left({\left(e^{z}\right)}^{a}\right)}}^{b}\right) \]

   8. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{\log \left({\left({\left(e^{z}\right)}^{a}\right)}^{b}\right)} \]
   9. Step-by-step derivation

      1. log-pow 37.7%

         \[\leadsto \color{blue}{b \cdot \log \left({\left(e^{z}\right)}^{a}\right)} \]

      2. log-pow 37.7%

         \[\leadsto b \cdot \color{blue}{\left(a \cdot \log \left(e^{z}\right)\right)} \]

      3. rem-log-exp 72.3%

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

   10. Simplified 72.3%

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

   if -3.8000000000000001e64 < b < 9.49999999999999927e-37 or 4.64999999999999973e160 < b < 4.59999999999999996e213

   1. Initial program 95.9%

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

      1. associate-+l+ 95.9%

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

      2. +-commutative 95.9%

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

      3. *-commutative 95.9%

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

      4. *-commutative 95.9%

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

      5. associate-*l* 98.6%

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

      6. distribute-rgt-out 98.6%

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

      7. fma-def 99.3%

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

      8. +-commutative 99.3%

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

      9. remove-double-neg 99.3%

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

      10. sub-neg 99.3%

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

      11. *-commutative 99.3%

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

      12. fma-neg 99.3%

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

      13. remove-double-neg 99.3%

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

      14. +-commutative 99.3%

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

      15. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 70.3%

      \[\leadsto \color{blue}{x + y \cdot z} \]

   if 9.49999999999999927e-37 < b < 3.09999999999999987e102

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. *-commutative 97.1%

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

      4. *-commutative 97.1%

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

      5. associate-*l* 97.0%

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

      6. distribute-rgt-out 99.8%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. *-commutative 99.8%

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

      12. fma-neg 99.8%

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

      13. remove-double-neg 99.8%

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

      14. +-commutative 99.8%

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

      15. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 70.5%

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

   if 3.09999999999999987e102 < b < 4.64999999999999973e160

   1. Initial program 92.9%

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

      1. associate-+l+ 92.9%

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

      2. +-commutative 92.9%

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

      3. *-commutative 92.9%

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

      4. *-commutative 92.9%

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

      5. associate-*l* 92.7%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 72.3%

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

      1. *-commutative 72.3%

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

   6. Simplified 72.3%

      \[\leadsto \color{blue}{a \cdot \left(z \cdot b\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+64}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-37}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{elif}\;b \leq 4.65 \cdot 10^{+160}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+213}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \]

Alternative 7: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \lor \neg \left(a \leq 5 \cdot 10^{-74}\right) \land \left(a \leq 6.7 \cdot 10^{+16} \lor \neg \left(a \leq 1.7 \cdot 10^{+78}\right)\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.1)
         (and (not (<= a 5e-74)) (or (<= a 6.7e+16) (not (<= a 1.7e+78)))))
   (* a (+ t (* z b)))
   (+ x (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.1) || (!(a <= 5e-74) && ((a <= 6.7e+16) || !(a <= 1.7e+78)))) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (y * z);
	}
	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 ((a <= (-1.1d0)) .or. (.not. (a <= 5d-74)) .and. (a <= 6.7d+16) .or. (.not. (a <= 1.7d+78))) then
        tmp = a * (t + (z * b))
    else
        tmp = x + (y * z)
    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 ((a <= -1.1) || (!(a <= 5e-74) && ((a <= 6.7e+16) || !(a <= 1.7e+78)))) {
		tmp = a * (t + (z * b));
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.1) or (not (a <= 5e-74) and ((a <= 6.7e+16) or not (a <= 1.7e+78))):
		tmp = a * (t + (z * b))
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.1) || (!(a <= 5e-74) && ((a <= 6.7e+16) || !(a <= 1.7e+78))))
		tmp = Float64(a * Float64(t + Float64(z * b)));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.1) || (~((a <= 5e-74)) && ((a <= 6.7e+16) || ~((a <= 1.7e+78)))))
		tmp = a * (t + (z * b));
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.1], And[N[Not[LessEqual[a, 5e-74]], $MachinePrecision], Or[LessEqual[a, 6.7e+16], N[Not[LessEqual[a, 1.7e+78]], $MachinePrecision]]]], N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001 or 4.99999999999999998e-74 < a < 6.7e16 or 1.70000000000000004e78 < a

   1. Initial program 89.7%

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

      1. associate-+l+ 89.7%

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

      2. +-commutative 89.7%

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

      3. *-commutative 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*l* 93.2%

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

      6. distribute-rgt-out 96.9%

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

      7. fma-def 97.7%

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

      8. +-commutative 97.7%

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

      9. remove-double-neg 97.7%

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

      10. sub-neg 97.7%

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

      11. *-commutative 97.7%

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

      12. fma-neg 97.7%

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

      13. remove-double-neg 97.7%

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

      14. +-commutative 97.7%

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

      15. fma-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in a around inf 75.3%

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

   if -1.1000000000000001 < a < 4.99999999999999998e-74 or 6.7e16 < a < 1.70000000000000004e78

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*l* 92.0%

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

      6. distribute-rgt-out 92.0%

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

      7. fma-def 92.0%

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

      8. +-commutative 92.0%

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

      9. remove-double-neg 92.0%

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

      10. sub-neg 92.0%

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

      11. *-commutative 92.0%

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

      12. fma-neg 92.0%

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

      13. remove-double-neg 92.0%

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

      14. +-commutative 92.0%

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

      15. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in a around 0 79.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \lor \neg \left(a \leq 5 \cdot 10^{-74}\right) \land \left(a \leq 6.7 \cdot 10^{+16} \lor \neg \left(a \leq 1.7 \cdot 10^{+78}\right)\right):\\ \;\;\;\;a \cdot \left(t + z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 8: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+63} \lor \neg \left(z \leq 2.55 \cdot 10^{-59}\right):\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -8.5e+63) (not (<= z 2.55e-59)))
   (* z (+ y (* a b)))
   (+ x (* a (+ t (* z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -8.5e+63) || !(z <= 2.55e-59)) {
		tmp = z * (y + (a * b));
	} else {
		tmp = x + (a * (t + (z * 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 ((z <= (-8.5d+63)) .or. (.not. (z <= 2.55d-59))) then
        tmp = z * (y + (a * b))
    else
        tmp = x + (a * (t + (z * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -8.5e+63) || !(z <= 2.55e-59)) {
		tmp = z * (y + (a * b));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -8.5e+63) or not (z <= 2.55e-59):
		tmp = z * (y + (a * b))
	else:
		tmp = x + (a * (t + (z * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -8.5e+63) || !(z <= 2.55e-59))
		tmp = Float64(z * Float64(y + Float64(a * b)));
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -8.5e+63) || ~((z <= 2.55e-59)))
		tmp = z * (y + (a * b));
	else
		tmp = x + (a * (t + (z * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -8.5e+63], N[Not[LessEqual[z, 2.55e-59]], $MachinePrecision]], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000004e63 or 2.5499999999999998e-59 < z

   1. Initial program 89.1%

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

      1. associate-+l+ 89.1%

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

      2. +-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*l* 86.9%

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

      6. distribute-rgt-out 89.9%

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

      7. fma-def 90.6%

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

      8. +-commutative 90.6%

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

      9. remove-double-neg 90.6%

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

      10. sub-neg 90.6%

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

      11. *-commutative 90.6%

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

      12. fma-neg 90.6%

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

      13. remove-double-neg 90.6%

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

      14. +-commutative 90.6%

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

      15. fma-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in z around inf 80.6%

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

   if -8.5000000000000004e63 < z < 2.5499999999999998e-59

   1. Initial program 99.1%

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

      1. associate-+l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*l* 99.1%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 91.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+63} \lor \neg \left(z \leq 2.55 \cdot 10^{-59}\right):\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 9: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+19} \lor \neg \left(z \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.8e+19) (not (<= z 5e-67)))
   (+ x (* z (+ y (* a b))))
   (+ x (* a (+ t (* z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.8e+19) || !(z <= 5e-67)) {
		tmp = x + (z * (y + (a * b)));
	} else {
		tmp = x + (a * (t + (z * 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 ((z <= (-2.8d+19)) .or. (.not. (z <= 5d-67))) then
        tmp = x + (z * (y + (a * b)))
    else
        tmp = x + (a * (t + (z * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.8e+19) || !(z <= 5e-67)) {
		tmp = x + (z * (y + (a * b)));
	} else {
		tmp = x + (a * (t + (z * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.8e+19) or not (z <= 5e-67):
		tmp = x + (z * (y + (a * b)))
	else:
		tmp = x + (a * (t + (z * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.8e+19) || !(z <= 5e-67))
		tmp = Float64(x + Float64(z * Float64(y + Float64(a * b))));
	else
		tmp = Float64(x + Float64(a * Float64(t + Float64(z * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.8e+19) || ~((z <= 5e-67)))
		tmp = x + (z * (y + (a * b)));
	else
		tmp = x + (a * (t + (z * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.8e+19], N[Not[LessEqual[z, 5e-67]], $MachinePrecision]], N[(x + N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(a * N[(t + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e19 or 4.9999999999999999e-67 < z

   1. Initial program 89.6%

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

      1. associate-+l+ 89.6%

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

      2. +-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*l* 87.5%

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

      6. distribute-rgt-out 90.3%

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

      7. fma-def 91.0%

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

      8. +-commutative 91.0%

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

      9. remove-double-neg 91.0%

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

      10. sub-neg 91.0%

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

      11. *-commutative 91.0%

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

      12. fma-neg 91.0%

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

      13. remove-double-neg 91.0%

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

      14. +-commutative 91.0%

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

      15. fma-def 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in t around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. associate-*r* 88.5%

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

      3. distribute-rgt-in 92.0%

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

   6. Simplified 92.0%

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

   if -2.8e19 < z < 4.9999999999999999e-67

   1. Initial program 99.1%

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

      1. associate-+l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*l* 99.0%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 92.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+19} \lor \neg \left(z \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;x + z \cdot \left(y + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \left(t + z \cdot b\right)\\ \end{array} \]

Alternative 10: 38.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+62}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.45e+62)
   (* y z)
   (if (<= z 2.55e-291)
     x
     (if (<= z 1.12e-155) (* t a) (if (<= z 4.5e-62) x (* y z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.45e+62) {
		tmp = y * z;
	} else if (z <= 2.55e-291) {
		tmp = x;
	} else if (z <= 1.12e-155) {
		tmp = t * a;
	} else if (z <= 4.5e-62) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	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 <= (-1.45d+62)) then
        tmp = y * z
    else if (z <= 2.55d-291) then
        tmp = x
    else if (z <= 1.12d-155) then
        tmp = t * a
    else if (z <= 4.5d-62) then
        tmp = x
    else
        tmp = y * z
    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 <= -1.45e+62) {
		tmp = y * z;
	} else if (z <= 2.55e-291) {
		tmp = x;
	} else if (z <= 1.12e-155) {
		tmp = t * a;
	} else if (z <= 4.5e-62) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.45e+62:
		tmp = y * z
	elif z <= 2.55e-291:
		tmp = x
	elif z <= 1.12e-155:
		tmp = t * a
	elif z <= 4.5e-62:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.45e+62)
		tmp = Float64(y * z);
	elseif (z <= 2.55e-291)
		tmp = x;
	elseif (z <= 1.12e-155)
		tmp = Float64(t * a);
	elseif (z <= 4.5e-62)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.45e+62)
		tmp = y * z;
	elseif (z <= 2.55e-291)
		tmp = x;
	elseif (z <= 1.12e-155)
		tmp = t * a;
	elseif (z <= 4.5e-62)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.45e+62], N[(y * z), $MachinePrecision], If[LessEqual[z, 2.55e-291], x, If[LessEqual[z, 1.12e-155], N[(t * a), $MachinePrecision], If[LessEqual[z, 4.5e-62], x, N[(y * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999992e62 or 4.50000000000000018e-62 < z

   1. Initial program 89.1%

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

      1. associate-+l+ 89.1%

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

      2. +-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*l* 86.9%

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

      6. distribute-rgt-out 89.9%

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

      7. fma-def 90.6%

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

      8. +-commutative 90.6%

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

      9. remove-double-neg 90.6%

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

      10. sub-neg 90.6%

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

      11. *-commutative 90.6%

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

      12. fma-neg 90.6%

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

      13. remove-double-neg 90.6%

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

      14. +-commutative 90.6%

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

      15. fma-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in y around inf 45.2%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -1.44999999999999992e62 < z < 2.55e-291 or 1.1200000000000001e-155 < z < 4.50000000000000018e-62

   1. Initial program 98.9%

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

      1. associate-+l+ 98.9%

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

      2. +-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*l* 98.8%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 48.2%

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

   if 2.55e-291 < z < 1.1200000000000001e-155

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 59.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+62}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-155}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 11: 57.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+101}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.2e+101) (* y z) (if (<= y 1.3e+73) (+ x (* t a)) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+101) {
		tmp = y * z;
	} else if (y <= 1.3e+73) {
		tmp = x + (t * a);
	} else {
		tmp = y * z;
	}
	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 <= (-6.2d+101)) then
        tmp = y * z
    else if (y <= 1.3d+73) then
        tmp = x + (t * a)
    else
        tmp = y * z
    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 <= -6.2e+101) {
		tmp = y * z;
	} else if (y <= 1.3e+73) {
		tmp = x + (t * a);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.2e+101:
		tmp = y * z
	elif y <= 1.3e+73:
		tmp = x + (t * a)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.2e+101)
		tmp = Float64(y * z);
	elseif (y <= 1.3e+73)
		tmp = Float64(x + Float64(t * a));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.2e+101)
		tmp = y * z;
	elseif (y <= 1.3e+73)
		tmp = x + (t * a);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.2e+101], N[(y * z), $MachinePrecision], If[LessEqual[y, 1.3e+73], N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999998e101 or 1.3e73 < y

   1. Initial program 91.4%

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

      1. associate-+l+ 91.4%

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

      2. +-commutative 91.4%

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

      3. *-commutative 91.4%

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

      4. *-commutative 91.4%

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

      5. associate-*l* 91.3%

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

      6. distribute-rgt-out 92.4%

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

      7. fma-def 93.5%

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

      8. +-commutative 93.5%

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

      9. remove-double-neg 93.5%

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

      10. sub-neg 93.5%

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

      11. *-commutative 93.5%

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

      12. fma-neg 93.5%

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

      13. remove-double-neg 93.5%

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

      14. +-commutative 93.5%

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

      15. fma-def 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around inf 60.8%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -6.19999999999999998e101 < y < 1.3e73

   1. Initial program 95.2%

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

      1. associate-+l+ 95.2%

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

      2. +-commutative 95.2%

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

      3. *-commutative 95.2%

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

      4. *-commutative 95.2%

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

      5. associate-*l* 93.4%

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

      6. distribute-rgt-out 95.8%

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

      7. fma-def 95.8%

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

      8. +-commutative 95.8%

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

      9. remove-double-neg 95.8%

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

      10. sub-neg 95.8%

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

      11. *-commutative 95.8%

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

      12. fma-neg 95.8%

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

      13. remove-double-neg 95.8%

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

      14. +-commutative 95.8%

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

      15. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in z around 0 58.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+101}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;x + t \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 12: 38.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+41}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.4e+41) (* t a) (if (<= a 1.7e-57) x (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.4e+41) {
		tmp = t * a;
	} else if (a <= 1.7e-57) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	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 (a <= (-6.4d+41)) then
        tmp = t * a
    else if (a <= 1.7d-57) then
        tmp = x
    else
        tmp = t * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.4e+41) {
		tmp = t * a;
	} else if (a <= 1.7e-57) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6.4e+41:
		tmp = t * a
	elif a <= 1.7e-57:
		tmp = x
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6.4e+41)
		tmp = Float64(t * a);
	elseif (a <= 1.7e-57)
		tmp = x;
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6.4e+41)
		tmp = t * a;
	elseif (a <= 1.7e-57)
		tmp = x;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6.4e+41], N[(t * a), $MachinePrecision], If[LessEqual[a, 1.7e-57], x, N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.40000000000000019e41 or 1.70000000000000008e-57 < a

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. +-commutative 89.4%

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

      3. *-commutative 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*l* 93.0%

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

      6. distribute-rgt-out 96.8%

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

      7. fma-def 97.6%

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

      8. +-commutative 97.6%

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

      9. remove-double-neg 97.6%

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

      10. sub-neg 97.6%

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

      11. *-commutative 97.6%

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

      12. fma-neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. +-commutative 97.6%

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

      15. fma-def 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t around inf 35.0%

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

   if -6.40000000000000019e41 < a < 1.70000000000000008e-57

   1. Initial program 98.4%

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

      1. associate-+l+ 98.4%

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

      2. +-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. *-commutative 98.4%

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

      5. associate-*l* 92.3%

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

      6. distribute-rgt-out 92.3%

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

      7. fma-def 92.3%

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

      8. +-commutative 92.3%

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

      9. remove-double-neg 92.3%

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

      10. sub-neg 92.3%

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

      11. *-commutative 92.3%

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

      12. fma-neg 92.3%

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

      13. remove-double-neg 92.3%

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

      14. +-commutative 92.3%

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

      15. fma-def 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around inf 40.9%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+41}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]

Alternative 13: 26.3% accurate, 15.0× 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 93.8%

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

   1. associate-+l+ 93.8%

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

   2. +-commutative 93.8%

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

   3. *-commutative 93.8%

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

   4. *-commutative 93.8%

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

   5. associate-*l* 92.6%

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

   6. distribute-rgt-out 94.6%

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

   7. fma-def 95.0%

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

   8. +-commutative 95.0%

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

   9. remove-double-neg 95.0%

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

   10. sub-neg 95.0%

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

   11. *-commutative 95.0%

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

   12. fma-neg 95.0%

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

   13. remove-double-neg 95.0%

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

   14. +-commutative 95.0%

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

   15. fma-def 95.0%

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

3. Simplified 95.0%

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

4. Taylor expanded in x around inf 27.6%

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

5. Final simplification 27.6%

   \[\leadsto x \]

Developer target: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = 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 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    else
        tmp = 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 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000.0) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))