Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.5% → 98.8%

Time: 9.7s

Alternatives: 16

Speedup: 1.0×

• 33.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - b \cdot \frac{a}{4}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (fma t (/ z 16.0) (fma x y (- c (* b (/ a 4.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(t, (z / 16.0), fma(x, y, (c - (b * (a / 4.0)))));
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(t, Float64(z / 16.0), fma(x, y, Float64(c - Float64(b * Float64(a / 4.0)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(t * N[(z / 16.0), $MachinePrecision] + N[(x * y + N[(c - N[(b * N[(a / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. associate-+l- 97.3%

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

   2. +-commutative 97.3%

      \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   3. associate--l+ 97.3%

      \[\leadsto \color{blue}{\frac{z \cdot t}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

   4. associate-*l/ 97.7%

      \[\leadsto \color{blue}{\frac{z}{16} \cdot t} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   5. *-commutative 97.7%

      \[\leadsto \color{blue}{t \cdot \frac{z}{16}} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   6. fma-def 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{z}{16}, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

   7. fma-neg 99.2%

      \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{\mathsf{fma}\left(x, y, -\left(\frac{a \cdot b}{4} - c\right)\right)}\right) \]

   8. neg-sub0 99.2%

      \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, \color{blue}{0 - \left(\frac{a \cdot b}{4} - c\right)}\right)\right) \]

   9. associate-+l- 99.2%

      \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, \color{blue}{\left(0 - \frac{a \cdot b}{4}\right) + c}\right)\right) \]

   10. neg-sub0 99.2%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, \color{blue}{\left(-\frac{a \cdot b}{4}\right)} + c\right)\right) \]

   11. +-commutative 99.2%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, \color{blue}{c + \left(-\frac{a \cdot b}{4}\right)}\right)\right) \]

   12. unsub-neg 99.2%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, \color{blue}{c - \frac{a \cdot b}{4}}\right)\right) \]

   13. *-commutative 99.2%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - \frac{\color{blue}{b \cdot a}}{4}\right)\right) \]

   14. associate-*r/ 99.2%

       \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - \color{blue}{b \cdot \frac{a}{4}}\right)\right) \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - b \cdot \frac{a}{4}\right)\right)} \]

4. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - b \cdot \frac{a}{4}\right)\right) \]

Alternative 2: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, t \cdot \frac{z}{16}\right) + \left(c - b \cdot \frac{a}{4}\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (fma x y (* t (/ z 16.0))) (- c (* b (/ a 4.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(x, y, (t * (z / 16.0))) + (c - (b * (a / 4.0)));
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(fma(x, y, Float64(t * Float64(z / 16.0))) + Float64(c - Float64(b * Float64(a / 4.0))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(x * y + N[(t * N[(z / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c - N[(b * N[(a / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. associate-+l- 97.3%

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

   2. fma-def 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   3. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

   4. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

4. Final simplification 98.0%

   \[\leadsto \mathsf{fma}\left(x, y, t \cdot \frac{z}{16}\right) + \left(c - b \cdot \frac{a}{4}\right) \]

Alternative 3: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ t_2 := c + t_1\\ t_3 := c + x \cdot y\\ t_4 := t_1 + \left(b \cdot a\right) \cdot -0.25\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-223}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-30}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+105}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z)))
        (t_2 (+ c t_1))
        (t_3 (+ c (* x y)))
        (t_4 (+ t_1 (* (* b a) -0.25))))
   (if (<= y -4.8e-15)
     t_3
     (if (<= y -3.4e-227)
       t_2
       (if (<= y 4.8e-223)
         t_4
         (if (<= y 2.45e-197)
           t_2
           (if (<= y 1.2e-30)
             t_4
             (if (<= y 1.25e+81)
               t_2
               (if (<= y 8.4e+105)
                 (+ c (* b (* a -0.25)))
                 (if (<= y 5.4e+163) t_3 (- (* x y) (* (* b a) 0.25))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double t_2 = c + t_1;
	double t_3 = c + (x * y);
	double t_4 = t_1 + ((b * a) * -0.25);
	double tmp;
	if (y <= -4.8e-15) {
		tmp = t_3;
	} else if (y <= -3.4e-227) {
		tmp = t_2;
	} else if (y <= 4.8e-223) {
		tmp = t_4;
	} else if (y <= 2.45e-197) {
		tmp = t_2;
	} else if (y <= 1.2e-30) {
		tmp = t_4;
	} else if (y <= 1.25e+81) {
		tmp = t_2;
	} else if (y <= 8.4e+105) {
		tmp = c + (b * (a * -0.25));
	} else if (y <= 5.4e+163) {
		tmp = t_3;
	} else {
		tmp = (x * y) - ((b * a) * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = 0.0625d0 * (t * z)
    t_2 = c + t_1
    t_3 = c + (x * y)
    t_4 = t_1 + ((b * a) * (-0.25d0))
    if (y <= (-4.8d-15)) then
        tmp = t_3
    else if (y <= (-3.4d-227)) then
        tmp = t_2
    else if (y <= 4.8d-223) then
        tmp = t_4
    else if (y <= 2.45d-197) then
        tmp = t_2
    else if (y <= 1.2d-30) then
        tmp = t_4
    else if (y <= 1.25d+81) then
        tmp = t_2
    else if (y <= 8.4d+105) then
        tmp = c + (b * (a * (-0.25d0)))
    else if (y <= 5.4d+163) then
        tmp = t_3
    else
        tmp = (x * y) - ((b * a) * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double t_2 = c + t_1;
	double t_3 = c + (x * y);
	double t_4 = t_1 + ((b * a) * -0.25);
	double tmp;
	if (y <= -4.8e-15) {
		tmp = t_3;
	} else if (y <= -3.4e-227) {
		tmp = t_2;
	} else if (y <= 4.8e-223) {
		tmp = t_4;
	} else if (y <= 2.45e-197) {
		tmp = t_2;
	} else if (y <= 1.2e-30) {
		tmp = t_4;
	} else if (y <= 1.25e+81) {
		tmp = t_2;
	} else if (y <= 8.4e+105) {
		tmp = c + (b * (a * -0.25));
	} else if (y <= 5.4e+163) {
		tmp = t_3;
	} else {
		tmp = (x * y) - ((b * a) * 0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (t * z)
	t_2 = c + t_1
	t_3 = c + (x * y)
	t_4 = t_1 + ((b * a) * -0.25)
	tmp = 0
	if y <= -4.8e-15:
		tmp = t_3
	elif y <= -3.4e-227:
		tmp = t_2
	elif y <= 4.8e-223:
		tmp = t_4
	elif y <= 2.45e-197:
		tmp = t_2
	elif y <= 1.2e-30:
		tmp = t_4
	elif y <= 1.25e+81:
		tmp = t_2
	elif y <= 8.4e+105:
		tmp = c + (b * (a * -0.25))
	elif y <= 5.4e+163:
		tmp = t_3
	else:
		tmp = (x * y) - ((b * a) * 0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	t_2 = Float64(c + t_1)
	t_3 = Float64(c + Float64(x * y))
	t_4 = Float64(t_1 + Float64(Float64(b * a) * -0.25))
	tmp = 0.0
	if (y <= -4.8e-15)
		tmp = t_3;
	elseif (y <= -3.4e-227)
		tmp = t_2;
	elseif (y <= 4.8e-223)
		tmp = t_4;
	elseif (y <= 2.45e-197)
		tmp = t_2;
	elseif (y <= 1.2e-30)
		tmp = t_4;
	elseif (y <= 1.25e+81)
		tmp = t_2;
	elseif (y <= 8.4e+105)
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	elseif (y <= 5.4e+163)
		tmp = t_3;
	else
		tmp = Float64(Float64(x * y) - Float64(Float64(b * a) * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 0.0625 * (t * z);
	t_2 = c + t_1;
	t_3 = c + (x * y);
	t_4 = t_1 + ((b * a) * -0.25);
	tmp = 0.0;
	if (y <= -4.8e-15)
		tmp = t_3;
	elseif (y <= -3.4e-227)
		tmp = t_2;
	elseif (y <= 4.8e-223)
		tmp = t_4;
	elseif (y <= 2.45e-197)
		tmp = t_2;
	elseif (y <= 1.2e-30)
		tmp = t_4;
	elseif (y <= 1.25e+81)
		tmp = t_2;
	elseif (y <= 8.4e+105)
		tmp = c + (b * (a * -0.25));
	elseif (y <= 5.4e+163)
		tmp = t_3;
	else
		tmp = (x * y) - ((b * a) * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 + N[(N[(b * a), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e-15], t$95$3, If[LessEqual[y, -3.4e-227], t$95$2, If[LessEqual[y, 4.8e-223], t$95$4, If[LessEqual[y, 2.45e-197], t$95$2, If[LessEqual[y, 1.2e-30], t$95$4, If[LessEqual[y, 1.25e+81], t$95$2, If[LessEqual[y, 8.4e+105], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e+163], t$95$3, N[(N[(x * y), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.7999999999999999e-15 or 8.4000000000000004e105 < y < 5.39999999999999998e163

   1. Initial program 94.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf 64.1%

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

   if -4.7999999999999999e-15 < y < -3.39999999999999979e-227 or 4.79999999999999971e-223 < y < 2.4500000000000001e-197 or 1.19999999999999992e-30 < y < 1.25e81

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf 60.7%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} + c \]

   if -3.39999999999999979e-227 < y < 4.79999999999999971e-223 or 2.4500000000000001e-197 < y < 1.19999999999999992e-30

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in x around 0 95.5%

      \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - 0.25 \cdot \left(a \cdot b\right)} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 95.5%

         \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) + \left(-0.25\right) \cdot \left(a \cdot b\right)} \]

      2. +-commutative 95.5%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \left(t \cdot z\right) + c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      3. *-commutative 95.5%

         \[\leadsto \left(\color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      4. *-commutative 95.5%

         \[\leadsto \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625 + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      5. associate-*l* 95.5%

         \[\leadsto \left(\color{blue}{z \cdot \left(t \cdot 0.0625\right)} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      6. fma-udef 95.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot 0.0625, c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      7. metadata-eval 95.5%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{-0.25} \cdot \left(a \cdot b\right) \]

      8. associate-*r* 95.5%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(-0.25 \cdot a\right) \cdot b} \]

      9. *-commutative 95.5%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(a \cdot -0.25\right)} \cdot b \]

      10. *-commutative 95.5%

          \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{b \cdot \left(a \cdot -0.25\right)} \]

      11. +-commutative 95.5%

          \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right) + \mathsf{fma}\left(z, t \cdot 0.0625, c\right)} \]

      12. fma-def 95.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, t \cdot 0.0625, c\right)\right)} \]

      13. *-commutative 95.5%

          \[\leadsto \mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, \color{blue}{0.0625 \cdot t}, c\right)\right) \]

   6. Simplified 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, 0.0625 \cdot t, c\right)\right)} \]

   7. Taylor expanded in c around 0 69.9%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right) + -0.25 \cdot \left(a \cdot b\right)} \]

   if 1.25e81 < y < 8.4000000000000004e105

   1. Initial program 66.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf 34.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(a \cdot b\right)} + c \]
   3. Step-by-step derivation

      1. associate-*r* 34.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot a\right) \cdot b} + c \]

      2. *-commutative 34.9%

         \[\leadsto \color{blue}{\left(a \cdot -0.25\right)} \cdot b + c \]

      3. *-commutative 34.9%

         \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} + c \]

   4. Simplified 34.9%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} + c \]

   if 5.39999999999999998e163 < y

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 95.7%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in z around 0 70.0%

      \[\leadsto \color{blue}{\left(c + y \cdot x\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   5. Taylor expanded in c around 0 65.7%

      \[\leadsto \color{blue}{y \cdot x - 0.25 \cdot \left(a \cdot b\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-227}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-223}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right) + \left(b \cdot a\right) \cdot -0.25\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-197}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-30}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right) + \left(b \cdot a\right) \cdot -0.25\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+105}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \end{array} \]

Alternative 4: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + 0.0625 \cdot \left(t \cdot z\right)\\ t_2 := x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -2 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-82}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* 0.0625 (* t z)))) (t_2 (- (* x y) (* (* b a) 0.25))))
   (if (<= (* b a) -1e+62)
     t_2
     (if (<= (* b a) -2e-270)
       t_1
       (if (<= (* b a) 5e-82)
         (+ c (* x y))
         (if (<= (* b a) 2e+132) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (0.0625 * (t * z));
	double t_2 = (x * y) - ((b * a) * 0.25);
	double tmp;
	if ((b * a) <= -1e+62) {
		tmp = t_2;
	} else if ((b * a) <= -2e-270) {
		tmp = t_1;
	} else if ((b * a) <= 5e-82) {
		tmp = c + (x * y);
	} else if ((b * a) <= 2e+132) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = c + (0.0625d0 * (t * z))
    t_2 = (x * y) - ((b * a) * 0.25d0)
    if ((b * a) <= (-1d+62)) then
        tmp = t_2
    else if ((b * a) <= (-2d-270)) then
        tmp = t_1
    else if ((b * a) <= 5d-82) then
        tmp = c + (x * y)
    else if ((b * a) <= 2d+132) 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 c) {
	double t_1 = c + (0.0625 * (t * z));
	double t_2 = (x * y) - ((b * a) * 0.25);
	double tmp;
	if ((b * a) <= -1e+62) {
		tmp = t_2;
	} else if ((b * a) <= -2e-270) {
		tmp = t_1;
	} else if ((b * a) <= 5e-82) {
		tmp = c + (x * y);
	} else if ((b * a) <= 2e+132) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (0.0625 * (t * z))
	t_2 = (x * y) - ((b * a) * 0.25)
	tmp = 0
	if (b * a) <= -1e+62:
		tmp = t_2
	elif (b * a) <= -2e-270:
		tmp = t_1
	elif (b * a) <= 5e-82:
		tmp = c + (x * y)
	elif (b * a) <= 2e+132:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(0.0625 * Float64(t * z)))
	t_2 = Float64(Float64(x * y) - Float64(Float64(b * a) * 0.25))
	tmp = 0.0
	if (Float64(b * a) <= -1e+62)
		tmp = t_2;
	elseif (Float64(b * a) <= -2e-270)
		tmp = t_1;
	elseif (Float64(b * a) <= 5e-82)
		tmp = Float64(c + Float64(x * y));
	elseif (Float64(b * a) <= 2e+132)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (0.0625 * (t * z));
	t_2 = (x * y) - ((b * a) * 0.25);
	tmp = 0.0;
	if ((b * a) <= -1e+62)
		tmp = t_2;
	elseif ((b * a) <= -2e-270)
		tmp = t_1;
	elseif ((b * a) <= 5e-82)
		tmp = c + (x * y);
	elseif ((b * a) <= 2e+132)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -1e+62], t$95$2, If[LessEqual[N[(b * a), $MachinePrecision], -2e-270], t$95$1, If[LessEqual[N[(b * a), $MachinePrecision], 5e-82], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2e+132], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.00000000000000004e62 or 1.99999999999999998e132 < (*.f64 a b)

   1. Initial program 94.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 94.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 94.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 95.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 95.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in z around 0 86.6%

      \[\leadsto \color{blue}{\left(c + y \cdot x\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   5. Taylor expanded in c around 0 79.7%

      \[\leadsto \color{blue}{y \cdot x - 0.25 \cdot \left(a \cdot b\right)} \]

   if -1.00000000000000004e62 < (*.f64 a b) < -2.0000000000000001e-270 or 4.9999999999999998e-82 < (*.f64 a b) < 1.99999999999999998e132

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf 68.6%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} + c \]

   if -2.0000000000000001e-270 < (*.f64 a b) < 4.9999999999999998e-82

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf 64.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{elif}\;b \cdot a \leq -2 \cdot 10^{-270}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-82}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+132}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \end{array} \]

Alternative 5: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot 0.25\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-20}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(c + t_2\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* b a) 0.25)) (t_2 (* 0.0625 (* t z))))
   (if (<= (* x y) -4e-20)
     (- (+ c (* x y)) t_1)
     (if (<= (* x y) 5e+131) (- (+ c t_2) t_1) (+ c (+ (* x y) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b * a) * 0.25;
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if ((x * y) <= -4e-20) {
		tmp = (c + (x * y)) - t_1;
	} else if ((x * y) <= 5e+131) {
		tmp = (c + t_2) - t_1;
	} else {
		tmp = c + ((x * y) + t_2);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (b * a) * 0.25d0
    t_2 = 0.0625d0 * (t * z)
    if ((x * y) <= (-4d-20)) then
        tmp = (c + (x * y)) - t_1
    else if ((x * y) <= 5d+131) then
        tmp = (c + t_2) - t_1
    else
        tmp = c + ((x * y) + 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 c) {
	double t_1 = (b * a) * 0.25;
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if ((x * y) <= -4e-20) {
		tmp = (c + (x * y)) - t_1;
	} else if ((x * y) <= 5e+131) {
		tmp = (c + t_2) - t_1;
	} else {
		tmp = c + ((x * y) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (b * a) * 0.25
	t_2 = 0.0625 * (t * z)
	tmp = 0
	if (x * y) <= -4e-20:
		tmp = (c + (x * y)) - t_1
	elif (x * y) <= 5e+131:
		tmp = (c + t_2) - t_1
	else:
		tmp = c + ((x * y) + t_2)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b * a) * 0.25)
	t_2 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(x * y) <= -4e-20)
		tmp = Float64(Float64(c + Float64(x * y)) - t_1);
	elseif (Float64(x * y) <= 5e+131)
		tmp = Float64(Float64(c + t_2) - t_1);
	else
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (b * a) * 0.25;
	t_2 = 0.0625 * (t * z);
	tmp = 0.0;
	if ((x * y) <= -4e-20)
		tmp = (c + (x * y)) - t_1;
	elseif ((x * y) <= 5e+131)
		tmp = (c + t_2) - t_1;
	else
		tmp = c + ((x * y) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -4e-20], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+131], N[(N[(c + t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999978e-20

   1. Initial program 94.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 94.2%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 94.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 94.2%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 94.2%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in z around 0 85.5%

      \[\leadsto \color{blue}{\left(c + y \cdot x\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   if -3.99999999999999978e-20 < (*.f64 x y) < 4.99999999999999995e131

   1. Initial program 99.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 99.4%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in x around 0 95.4%

      \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   if 4.99999999999999995e131 < (*.f64 x y)

   1. Initial program 95.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around 0 92.9%

      \[\leadsto \color{blue}{\left(y \cdot x + 0.0625 \cdot \left(t \cdot z\right)\right)} + c \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-20}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]

Alternative 6: 37.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ t_2 := b \cdot \left(a \cdot -0.25\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{+193}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))) (t_2 (* b (* a -0.25))))
   (if (<= x -3e+193)
     (* x y)
     (if (<= x -1.15e+143)
       t_1
       (if (<= x -1.2e+96)
         (* x y)
         (if (<= x -7.8e-94)
           t_2
           (if (<= x -1.05e-121)
             t_1
             (if (<= x -2.7e-248) t_2 (if (<= x 5.5e+39) t_1 (* x y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double t_2 = b * (a * -0.25);
	double tmp;
	if (x <= -3e+193) {
		tmp = x * y;
	} else if (x <= -1.15e+143) {
		tmp = t_1;
	} else if (x <= -1.2e+96) {
		tmp = x * y;
	} else if (x <= -7.8e-94) {
		tmp = t_2;
	} else if (x <= -1.05e-121) {
		tmp = t_1;
	} else if (x <= -2.7e-248) {
		tmp = t_2;
	} else if (x <= 5.5e+39) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.0625d0 * (t * z)
    t_2 = b * (a * (-0.25d0))
    if (x <= (-3d+193)) then
        tmp = x * y
    else if (x <= (-1.15d+143)) then
        tmp = t_1
    else if (x <= (-1.2d+96)) then
        tmp = x * y
    else if (x <= (-7.8d-94)) then
        tmp = t_2
    else if (x <= (-1.05d-121)) then
        tmp = t_1
    else if (x <= (-2.7d-248)) then
        tmp = t_2
    else if (x <= 5.5d+39) then
        tmp = t_1
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double t_2 = b * (a * -0.25);
	double tmp;
	if (x <= -3e+193) {
		tmp = x * y;
	} else if (x <= -1.15e+143) {
		tmp = t_1;
	} else if (x <= -1.2e+96) {
		tmp = x * y;
	} else if (x <= -7.8e-94) {
		tmp = t_2;
	} else if (x <= -1.05e-121) {
		tmp = t_1;
	} else if (x <= -2.7e-248) {
		tmp = t_2;
	} else if (x <= 5.5e+39) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (t * z)
	t_2 = b * (a * -0.25)
	tmp = 0
	if x <= -3e+193:
		tmp = x * y
	elif x <= -1.15e+143:
		tmp = t_1
	elif x <= -1.2e+96:
		tmp = x * y
	elif x <= -7.8e-94:
		tmp = t_2
	elif x <= -1.05e-121:
		tmp = t_1
	elif x <= -2.7e-248:
		tmp = t_2
	elif x <= 5.5e+39:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	t_2 = Float64(b * Float64(a * -0.25))
	tmp = 0.0
	if (x <= -3e+193)
		tmp = Float64(x * y);
	elseif (x <= -1.15e+143)
		tmp = t_1;
	elseif (x <= -1.2e+96)
		tmp = Float64(x * y);
	elseif (x <= -7.8e-94)
		tmp = t_2;
	elseif (x <= -1.05e-121)
		tmp = t_1;
	elseif (x <= -2.7e-248)
		tmp = t_2;
	elseif (x <= 5.5e+39)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 0.0625 * (t * z);
	t_2 = b * (a * -0.25);
	tmp = 0.0;
	if (x <= -3e+193)
		tmp = x * y;
	elseif (x <= -1.15e+143)
		tmp = t_1;
	elseif (x <= -1.2e+96)
		tmp = x * y;
	elseif (x <= -7.8e-94)
		tmp = t_2;
	elseif (x <= -1.05e-121)
		tmp = t_1;
	elseif (x <= -2.7e-248)
		tmp = t_2;
	elseif (x <= 5.5e+39)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+193], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.15e+143], t$95$1, If[LessEqual[x, -1.2e+96], N[(x * y), $MachinePrecision], If[LessEqual[x, -7.8e-94], t$95$2, If[LessEqual[x, -1.05e-121], t$95$1, If[LessEqual[x, -2.7e-248], t$95$2, If[LessEqual[x, 5.5e+39], t$95$1, N[(x * y), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e193 or -1.15e143 < x < -1.19999999999999996e96 or 5.4999999999999997e39 < x

   1. Initial program 93.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 93.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 94.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 94.2%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 94.2%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 93.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 93.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 93.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 93.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in x around inf 60.3%

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

   if -3e193 < x < -1.15e143 or -7.8000000000000004e-94 < x < -1.0499999999999999e-121 or -2.7000000000000001e-248 < x < 5.4999999999999997e39

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in t around inf 48.1%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]

   if -1.19999999999999996e96 < x < -7.8000000000000004e-94 or -1.0499999999999999e-121 < x < -2.7000000000000001e-248

   1. Initial program 98.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 98.7%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in a around inf 28.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative 28.1%

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot -0.25} \]

      2. *-commutative 28.1%

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot -0.25 \]

      3. associate-*r* 28.1%

         \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} \]

   8. Simplified 28.1%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+193}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+143}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-94}:\\ \;\;\;\;b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-121}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-248}:\\ \;\;\;\;b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 60.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + 0.0625 \cdot \left(t \cdot z\right)\\ t_2 := c + x \cdot y\\ t_3 := c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-303}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* 0.0625 (* t z))))
        (t_2 (+ c (* x y)))
        (t_3 (+ c (* b (* a -0.25)))))
   (if (<= y -9e-15)
     t_2
     (if (<= y -1.65e-244)
       t_1
       (if (<= y -5.6e-303)
         t_3
         (if (<= y 5.5e-241)
           t_1
           (if (<= y 1.65e-102) t_3 (if (<= y 1.55e+60) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (0.0625 * (t * z));
	double t_2 = c + (x * y);
	double t_3 = c + (b * (a * -0.25));
	double tmp;
	if (y <= -9e-15) {
		tmp = t_2;
	} else if (y <= -1.65e-244) {
		tmp = t_1;
	} else if (y <= -5.6e-303) {
		tmp = t_3;
	} else if (y <= 5.5e-241) {
		tmp = t_1;
	} else if (y <= 1.65e-102) {
		tmp = t_3;
	} else if (y <= 1.55e+60) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = c + (0.0625d0 * (t * z))
    t_2 = c + (x * y)
    t_3 = c + (b * (a * (-0.25d0)))
    if (y <= (-9d-15)) then
        tmp = t_2
    else if (y <= (-1.65d-244)) then
        tmp = t_1
    else if (y <= (-5.6d-303)) then
        tmp = t_3
    else if (y <= 5.5d-241) then
        tmp = t_1
    else if (y <= 1.65d-102) then
        tmp = t_3
    else if (y <= 1.55d+60) 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 c) {
	double t_1 = c + (0.0625 * (t * z));
	double t_2 = c + (x * y);
	double t_3 = c + (b * (a * -0.25));
	double tmp;
	if (y <= -9e-15) {
		tmp = t_2;
	} else if (y <= -1.65e-244) {
		tmp = t_1;
	} else if (y <= -5.6e-303) {
		tmp = t_3;
	} else if (y <= 5.5e-241) {
		tmp = t_1;
	} else if (y <= 1.65e-102) {
		tmp = t_3;
	} else if (y <= 1.55e+60) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (0.0625 * (t * z))
	t_2 = c + (x * y)
	t_3 = c + (b * (a * -0.25))
	tmp = 0
	if y <= -9e-15:
		tmp = t_2
	elif y <= -1.65e-244:
		tmp = t_1
	elif y <= -5.6e-303:
		tmp = t_3
	elif y <= 5.5e-241:
		tmp = t_1
	elif y <= 1.65e-102:
		tmp = t_3
	elif y <= 1.55e+60:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(0.0625 * Float64(t * z)))
	t_2 = Float64(c + Float64(x * y))
	t_3 = Float64(c + Float64(b * Float64(a * -0.25)))
	tmp = 0.0
	if (y <= -9e-15)
		tmp = t_2;
	elseif (y <= -1.65e-244)
		tmp = t_1;
	elseif (y <= -5.6e-303)
		tmp = t_3;
	elseif (y <= 5.5e-241)
		tmp = t_1;
	elseif (y <= 1.65e-102)
		tmp = t_3;
	elseif (y <= 1.55e+60)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (0.0625 * (t * z));
	t_2 = c + (x * y);
	t_3 = c + (b * (a * -0.25));
	tmp = 0.0;
	if (y <= -9e-15)
		tmp = t_2;
	elseif (y <= -1.65e-244)
		tmp = t_1;
	elseif (y <= -5.6e-303)
		tmp = t_3;
	elseif (y <= 5.5e-241)
		tmp = t_1;
	elseif (y <= 1.65e-102)
		tmp = t_3;
	elseif (y <= 1.55e+60)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e-15], t$95$2, If[LessEqual[y, -1.65e-244], t$95$1, If[LessEqual[y, -5.6e-303], t$95$3, If[LessEqual[y, 5.5e-241], t$95$1, If[LessEqual[y, 1.65e-102], t$95$3, If[LessEqual[y, 1.55e+60], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999995e-15 or 1.55e60 < y

   1. Initial program 94.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf 63.4%

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

   if -8.9999999999999995e-15 < y < -1.65000000000000013e-244 or -5.6e-303 < y < 5.4999999999999998e-241 or 1.65e-102 < y < 1.55e60

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf 62.7%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} + c \]

   if -1.65000000000000013e-244 < y < -5.6e-303 or 5.4999999999999998e-241 < y < 1.65e-102

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf 62.5%

      \[\leadsto \color{blue}{-0.25 \cdot \left(a \cdot b\right)} + c \]
   3. Step-by-step derivation

      1. associate-*r* 62.5%

         \[\leadsto \color{blue}{\left(-0.25 \cdot a\right) \cdot b} + c \]

      2. *-commutative 62.5%

         \[\leadsto \color{blue}{\left(a \cdot -0.25\right)} \cdot b + c \]

      3. *-commutative 62.5%

         \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} + c \]

   4. Simplified 62.5%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right)} + c \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-15}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-244}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-303}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-241}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-102}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 8: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+14} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\left(c + x \cdot y\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* b a) -5e+14) (not (<= (* b a) 2e+132)))
   (- (+ c (* x y)) (* (* b a) 0.25))
   (+ c (+ (* x y) (* 0.0625 (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((b * a) <= -5e+14) || !((b * a) <= 2e+132)) {
		tmp = (c + (x * y)) - ((b * a) * 0.25);
	} else {
		tmp = c + ((x * y) + (0.0625 * (t * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: tmp
    if (((b * a) <= (-5d+14)) .or. (.not. ((b * a) <= 2d+132))) then
        tmp = (c + (x * y)) - ((b * a) * 0.25d0)
    else
        tmp = c + ((x * y) + (0.0625d0 * (t * 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 c) {
	double tmp;
	if (((b * a) <= -5e+14) || !((b * a) <= 2e+132)) {
		tmp = (c + (x * y)) - ((b * a) * 0.25);
	} else {
		tmp = c + ((x * y) + (0.0625 * (t * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((b * a) <= -5e+14) or not ((b * a) <= 2e+132):
		tmp = (c + (x * y)) - ((b * a) * 0.25)
	else:
		tmp = c + ((x * y) + (0.0625 * (t * z)))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(b * a) <= -5e+14) || !(Float64(b * a) <= 2e+132))
		tmp = Float64(Float64(c + Float64(x * y)) - Float64(Float64(b * a) * 0.25));
	else
		tmp = Float64(c + Float64(Float64(x * y) + Float64(0.0625 * Float64(t * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((b * a) <= -5e+14) || ~(((b * a) <= 2e+132)))
		tmp = (c + (x * y)) - ((b * a) * 0.25);
	else
		tmp = c + ((x * y) + (0.0625 * (t * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(b * a), $MachinePrecision], -5e+14], N[Not[LessEqual[N[(b * a), $MachinePrecision], 2e+132]], $MachinePrecision]], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(x * y), $MachinePrecision] + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5e14 or 1.99999999999999998e132 < (*.f64 a b)

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 95.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 95.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 95.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 95.9%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in z around 0 85.4%

      \[\leadsto \color{blue}{\left(c + y \cdot x\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   if -5e14 < (*.f64 a b) < 1.99999999999999998e132

   1. Initial program 98.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around 0 95.1%

      \[\leadsto \color{blue}{\left(y \cdot x + 0.0625 \cdot \left(t \cdot z\right)\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+14} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\left(c + x \cdot y\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]

Alternative 9: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;b \cdot a \leq -1.1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+141}:\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(b \cdot a\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* b a) -1.1e+62)
     (- (* x y) (* (* b a) 0.25))
     (if (<= (* b a) 2e+141)
       (+ c (+ (* x y) t_1))
       (+ t_1 (* (* b a) -0.25))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((b * a) <= -1.1e+62) {
		tmp = (x * y) - ((b * a) * 0.25);
	} else if ((b * a) <= 2e+141) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = t_1 + ((b * a) * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0625d0 * (t * z)
    if ((b * a) <= (-1.1d+62)) then
        tmp = (x * y) - ((b * a) * 0.25d0)
    else if ((b * a) <= 2d+141) then
        tmp = c + ((x * y) + t_1)
    else
        tmp = t_1 + ((b * a) * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((b * a) <= -1.1e+62) {
		tmp = (x * y) - ((b * a) * 0.25);
	} else if ((b * a) <= 2e+141) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = t_1 + ((b * a) * -0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (t * z)
	tmp = 0
	if (b * a) <= -1.1e+62:
		tmp = (x * y) - ((b * a) * 0.25)
	elif (b * a) <= 2e+141:
		tmp = c + ((x * y) + t_1)
	else:
		tmp = t_1 + ((b * a) * -0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(b * a) <= -1.1e+62)
		tmp = Float64(Float64(x * y) - Float64(Float64(b * a) * 0.25));
	elseif (Float64(b * a) <= 2e+141)
		tmp = Float64(c + Float64(Float64(x * y) + t_1));
	else
		tmp = Float64(t_1 + Float64(Float64(b * a) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 0.0625 * (t * z);
	tmp = 0.0;
	if ((b * a) <= -1.1e+62)
		tmp = (x * y) - ((b * a) * 0.25);
	elseif ((b * a) <= 2e+141)
		tmp = c + ((x * y) + t_1);
	else
		tmp = t_1 + ((b * a) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * a), $MachinePrecision], -1.1e+62], N[(N[(x * y), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * a), $MachinePrecision], 2e+141], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(b * a), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.10000000000000007e62

   1. Initial program 91.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 91.4%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 91.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 92.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 92.9%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in z around 0 85.9%

      \[\leadsto \color{blue}{\left(c + y \cdot x\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   5. Taylor expanded in c around 0 79.4%

      \[\leadsto \color{blue}{y \cdot x - 0.25 \cdot \left(a \cdot b\right)} \]

   if -1.10000000000000007e62 < (*.f64 a b) < 2.00000000000000003e141

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around 0 93.4%

      \[\leadsto \color{blue}{\left(y \cdot x + 0.0625 \cdot \left(t \cdot z\right)\right)} + c \]

   if 2.00000000000000003e141 < (*.f64 a b)

   1. Initial program 99.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - 0.25 \cdot \left(a \cdot b\right)} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 95.8%

         \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) + \left(-0.25\right) \cdot \left(a \cdot b\right)} \]

      2. +-commutative 95.8%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \left(t \cdot z\right) + c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      3. *-commutative 95.8%

         \[\leadsto \left(\color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      4. *-commutative 95.8%

         \[\leadsto \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625 + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      5. associate-*l* 95.8%

         \[\leadsto \left(\color{blue}{z \cdot \left(t \cdot 0.0625\right)} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      6. fma-udef 95.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot 0.0625, c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      7. metadata-eval 95.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{-0.25} \cdot \left(a \cdot b\right) \]

      8. associate-*r* 95.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(-0.25 \cdot a\right) \cdot b} \]

      9. *-commutative 95.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(a \cdot -0.25\right)} \cdot b \]

      10. *-commutative 95.8%

          \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{b \cdot \left(a \cdot -0.25\right)} \]

      11. +-commutative 95.8%

          \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right) + \mathsf{fma}\left(z, t \cdot 0.0625, c\right)} \]

      12. fma-def 95.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, t \cdot 0.0625, c\right)\right)} \]

      13. *-commutative 95.9%

          \[\leadsto \mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, \color{blue}{0.0625 \cdot t}, c\right)\right) \]

   6. Simplified 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, 0.0625 \cdot t, c\right)\right)} \]

   7. Taylor expanded in c around 0 86.6%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right) + -0.25 \cdot \left(a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -1.1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+141}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right) + \left(b \cdot a\right) \cdot -0.25\\ \end{array} \]

Alternative 10: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c + \left(\left(\frac{t \cdot z}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ c (- (+ (/ (* t z) 16.0) (* x y)) (/ (* b a) 4.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return c + ((((t * z) / 16.0) + (x * y)) - ((b * a) / 4.0));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    code = c + ((((t * z) / 16.0d0) + (x * y)) - ((b * a) / 4.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return c + ((((t * z) / 16.0) + (x * y)) - ((b * a) / 4.0));
}

Python

def code(x, y, z, t, a, b, c):
	return c + ((((t * z) / 16.0) + (x * y)) - ((b * a) / 4.0))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(c + Float64(Float64(Float64(Float64(t * z) / 16.0) + Float64(x * y)) - Float64(Float64(b * a) / 4.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = c + ((((t * z) / 16.0) + (x * y)) - ((b * a) / 4.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(c + N[(N[(N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Final simplification 97.3%

   \[\leadsto c + \left(\left(\frac{t \cdot z}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\right) \]

Alternative 11: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(t \cdot \left(z \cdot 0.0625\right) + x \cdot y\right) + \left(c - b \cdot \frac{a}{4}\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (+ (* t (* z 0.0625)) (* x y)) (- c (* b (/ a 4.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((t * (z * 0.0625)) + (x * y)) + (c - (b * (a / 4.0)));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    code = ((t * (z * 0.0625d0)) + (x * y)) + (c - (b * (a / 4.0d0)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((t * (z * 0.0625)) + (x * y)) + (c - (b * (a / 4.0)));
}

Python

def code(x, y, z, t, a, b, c):
	return ((t * (z * 0.0625)) + (x * y)) + (c - (b * (a / 4.0)))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(t * Float64(z * 0.0625)) + Float64(x * y)) + Float64(c - Float64(b * Float64(a / 4.0))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((t * (z * 0.0625)) + (x * y)) + (c - (b * (a / 4.0)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(c - N[(b * N[(a / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. associate-+l- 97.3%

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

   2. fma-def 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   3. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

   4. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
4. Step-by-step derivation

   1. fma-udef 97.7%

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   2. *-commutative 97.7%

      \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   3. div-inv 97.7%

      \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   4. metadata-eval 97.7%

      \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

5. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

6. Final simplification 97.7%

   \[\leadsto \left(t \cdot \left(z \cdot 0.0625\right) + x \cdot y\right) + \left(c - b \cdot \frac{a}{4}\right) \]

Alternative 12: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + x \cdot y\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ t_3 := c + t_2\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* x y))) (t_2 (* 0.0625 (* t z))) (t_3 (+ c t_2)))
   (if (<= t -5.5e-108)
     t_3
     (if (<= t 2.1e-30)
       t_1
       (if (<= t 7.1e-9) t_2 (if (<= t 6.6e+24) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (x * y);
	double t_2 = 0.0625 * (t * z);
	double t_3 = c + t_2;
	double tmp;
	if (t <= -5.5e-108) {
		tmp = t_3;
	} else if (t <= 2.1e-30) {
		tmp = t_1;
	} else if (t <= 7.1e-9) {
		tmp = t_2;
	} else if (t <= 6.6e+24) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = c + (x * y)
    t_2 = 0.0625d0 * (t * z)
    t_3 = c + t_2
    if (t <= (-5.5d-108)) then
        tmp = t_3
    else if (t <= 2.1d-30) then
        tmp = t_1
    else if (t <= 7.1d-9) then
        tmp = t_2
    else if (t <= 6.6d+24) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (x * y);
	double t_2 = 0.0625 * (t * z);
	double t_3 = c + t_2;
	double tmp;
	if (t <= -5.5e-108) {
		tmp = t_3;
	} else if (t <= 2.1e-30) {
		tmp = t_1;
	} else if (t <= 7.1e-9) {
		tmp = t_2;
	} else if (t <= 6.6e+24) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (x * y)
	t_2 = 0.0625 * (t * z)
	t_3 = c + t_2
	tmp = 0
	if t <= -5.5e-108:
		tmp = t_3
	elif t <= 2.1e-30:
		tmp = t_1
	elif t <= 7.1e-9:
		tmp = t_2
	elif t <= 6.6e+24:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(x * y))
	t_2 = Float64(0.0625 * Float64(t * z))
	t_3 = Float64(c + t_2)
	tmp = 0.0
	if (t <= -5.5e-108)
		tmp = t_3;
	elseif (t <= 2.1e-30)
		tmp = t_1;
	elseif (t <= 7.1e-9)
		tmp = t_2;
	elseif (t <= 6.6e+24)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (x * y);
	t_2 = 0.0625 * (t * z);
	t_3 = c + t_2;
	tmp = 0.0;
	if (t <= -5.5e-108)
		tmp = t_3;
	elseif (t <= 2.1e-30)
		tmp = t_1;
	elseif (t <= 7.1e-9)
		tmp = t_2;
	elseif (t <= 6.6e+24)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(c + t$95$2), $MachinePrecision]}, If[LessEqual[t, -5.5e-108], t$95$3, If[LessEqual[t, 2.1e-30], t$95$1, If[LessEqual[t, 7.1e-9], t$95$2, If[LessEqual[t, 6.6e+24], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.50000000000000031e-108 or 6.5999999999999998e24 < t

   1. Initial program 97.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf 59.9%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} + c \]

   if -5.50000000000000031e-108 < t < 2.1000000000000002e-30 or 7.09999999999999988e-9 < t < 6.5999999999999998e24

   1. Initial program 97.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf 58.0%

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

   if 2.1000000000000002e-30 < t < 7.09999999999999988e-9

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in t around inf 22.5%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+24}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]

Alternative 13: 37.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+193}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+143} \lor \neg \left(x \leq -7.4 \cdot 10^{+82}\right) \land x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= x -1.45e+193)
   (* x y)
   (if (or (<= x -1.15e+143) (and (not (<= x -7.4e+82)) (<= x 2e+61)))
     (* 0.0625 (* t z))
     (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (x <= -1.45e+193) {
		tmp = x * y;
	} else if ((x <= -1.15e+143) || (!(x <= -7.4e+82) && (x <= 2e+61))) {
		tmp = 0.0625 * (t * z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: tmp
    if (x <= (-1.45d+193)) then
        tmp = x * y
    else if ((x <= (-1.15d+143)) .or. (.not. (x <= (-7.4d+82))) .and. (x <= 2d+61)) then
        tmp = 0.0625d0 * (t * z)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (x <= -1.45e+193) {
		tmp = x * y;
	} else if ((x <= -1.15e+143) || (!(x <= -7.4e+82) && (x <= 2e+61))) {
		tmp = 0.0625 * (t * z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if x <= -1.45e+193:
		tmp = x * y
	elif (x <= -1.15e+143) or (not (x <= -7.4e+82) and (x <= 2e+61)):
		tmp = 0.0625 * (t * z)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (x <= -1.45e+193)
		tmp = Float64(x * y);
	elseif ((x <= -1.15e+143) || (!(x <= -7.4e+82) && (x <= 2e+61)))
		tmp = Float64(0.0625 * Float64(t * z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (x <= -1.45e+193)
		tmp = x * y;
	elseif ((x <= -1.15e+143) || (~((x <= -7.4e+82)) && (x <= 2e+61)))
		tmp = 0.0625 * (t * z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[x, -1.45e+193], N[(x * y), $MachinePrecision], If[Or[LessEqual[x, -1.15e+143], And[N[Not[LessEqual[x, -7.4e+82]], $MachinePrecision], LessEqual[x, 2e+61]]], N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000007e193 or -1.15e143 < x < -7.4000000000000005e82 or 1.9999999999999999e61 < x

   1. Initial program 94.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 94.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 95.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 95.2%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 95.2%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 94.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 94.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 94.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 94.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 94.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in x around inf 59.0%

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

   if -1.45000000000000007e193 < x < -1.15e143 or -7.4000000000000005e82 < x < 1.9999999999999999e61

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 98.9%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 99.4%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 99.4%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 99.4%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 99.4%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 99.4%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in t around inf 41.2%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+193}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+143} \lor \neg \left(x \leq -7.4 \cdot 10^{+82}\right) \land x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 50.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + x \cdot y\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* x y))) (t_2 (* 0.0625 (* t z))))
   (if (<= t -5.4e-105)
     t_2
     (if (<= t 2.1e-30)
       t_1
       (if (<= t 7.4e-9) t_2 (if (<= t 9.5e+115) t_1 (* t (* z 0.0625))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (x * y);
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if (t <= -5.4e-105) {
		tmp = t_2;
	} else if (t <= 2.1e-30) {
		tmp = t_1;
	} else if (t <= 7.4e-9) {
		tmp = t_2;
	} else if (t <= 9.5e+115) {
		tmp = t_1;
	} else {
		tmp = t * (z * 0.0625);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = c + (x * y)
    t_2 = 0.0625d0 * (t * z)
    if (t <= (-5.4d-105)) then
        tmp = t_2
    else if (t <= 2.1d-30) then
        tmp = t_1
    else if (t <= 7.4d-9) then
        tmp = t_2
    else if (t <= 9.5d+115) then
        tmp = t_1
    else
        tmp = t * (z * 0.0625d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (x * y);
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if (t <= -5.4e-105) {
		tmp = t_2;
	} else if (t <= 2.1e-30) {
		tmp = t_1;
	} else if (t <= 7.4e-9) {
		tmp = t_2;
	} else if (t <= 9.5e+115) {
		tmp = t_1;
	} else {
		tmp = t * (z * 0.0625);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (x * y)
	t_2 = 0.0625 * (t * z)
	tmp = 0
	if t <= -5.4e-105:
		tmp = t_2
	elif t <= 2.1e-30:
		tmp = t_1
	elif t <= 7.4e-9:
		tmp = t_2
	elif t <= 9.5e+115:
		tmp = t_1
	else:
		tmp = t * (z * 0.0625)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(x * y))
	t_2 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (t <= -5.4e-105)
		tmp = t_2;
	elseif (t <= 2.1e-30)
		tmp = t_1;
	elseif (t <= 7.4e-9)
		tmp = t_2;
	elseif (t <= 9.5e+115)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(z * 0.0625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (x * y);
	t_2 = 0.0625 * (t * z);
	tmp = 0.0;
	if (t <= -5.4e-105)
		tmp = t_2;
	elseif (t <= 2.1e-30)
		tmp = t_1;
	elseif (t <= 7.4e-9)
		tmp = t_2;
	elseif (t <= 9.5e+115)
		tmp = t_1;
	else
		tmp = t * (z * 0.0625);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.4e-105], t$95$2, If[LessEqual[t, 2.1e-30], t$95$1, If[LessEqual[t, 7.4e-9], t$95$2, If[LessEqual[t, 9.5e+115], t$95$1, N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.39999999999999985e-105 or 2.1000000000000002e-30 < t < 7.4e-9

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 99.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 99.0%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 99.0%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in t around inf 43.7%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]

   if -5.39999999999999985e-105 < t < 2.1000000000000002e-30 or 7.4e-9 < t < 9.4999999999999997e115

   1. Initial program 97.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf 55.2%

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

   if 9.4999999999999997e115 < t

   1. Initial program 90.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 90.8%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 90.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 93.5%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 93.5%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in x around 0 72.0%

      \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - 0.25 \cdot \left(a \cdot b\right)} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 72.0%

         \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) + \left(-0.25\right) \cdot \left(a \cdot b\right)} \]

      2. +-commutative 72.0%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \left(t \cdot z\right) + c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      3. *-commutative 72.0%

         \[\leadsto \left(\color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      4. *-commutative 72.0%

         \[\leadsto \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625 + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      5. associate-*l* 74.8%

         \[\leadsto \left(\color{blue}{z \cdot \left(t \cdot 0.0625\right)} + c\right) + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      6. fma-udef 74.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot 0.0625, c\right)} + \left(-0.25\right) \cdot \left(a \cdot b\right) \]

      7. metadata-eval 74.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{-0.25} \cdot \left(a \cdot b\right) \]

      8. associate-*r* 74.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(-0.25 \cdot a\right) \cdot b} \]

      9. *-commutative 74.8%

         \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{\left(a \cdot -0.25\right)} \cdot b \]

      10. *-commutative 74.8%

          \[\leadsto \mathsf{fma}\left(z, t \cdot 0.0625, c\right) + \color{blue}{b \cdot \left(a \cdot -0.25\right)} \]

      11. +-commutative 74.8%

          \[\leadsto \color{blue}{b \cdot \left(a \cdot -0.25\right) + \mathsf{fma}\left(z, t \cdot 0.0625, c\right)} \]

      12. fma-def 74.8%

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, t \cdot 0.0625, c\right)\right)} \]

      13. *-commutative 74.8%

          \[\leadsto \mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, \color{blue}{0.0625 \cdot t}, c\right)\right) \]

   6. Simplified 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a \cdot -0.25, \mathsf{fma}\left(z, 0.0625 \cdot t, c\right)\right)} \]

   7. Taylor expanded in c around 0 65.5%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right) + -0.25 \cdot \left(a \cdot b\right)} \]

   8. Taylor expanded in t around inf 52.8%

      \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 55.5%

         \[\leadsto \color{blue}{\left(0.0625 \cdot t\right) \cdot z} \]

      2. *-commutative 55.5%

         \[\leadsto \color{blue}{\left(t \cdot 0.0625\right)} \cdot z \]

      3. associate-*r* 55.5%

         \[\leadsto \color{blue}{t \cdot \left(0.0625 \cdot z\right)} \]

   10. Simplified 55.5%

       \[\leadsto \color{blue}{t \cdot \left(0.0625 \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-105}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+115}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \]

Alternative 15: 37.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-25}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+41}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= y -2.8e-25) (* x y) (if (<= y 9e+41) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (y <= -2.8e-25) {
		tmp = x * y;
	} else if (y <= 9e+41) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: c
    real(8) :: tmp
    if (y <= (-2.8d-25)) then
        tmp = x * y
    else if (y <= 9d+41) then
        tmp = c
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (y <= -2.8e-25) {
		tmp = x * y;
	} else if (y <= 9e+41) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if y <= -2.8e-25:
		tmp = x * y
	elif y <= 9e+41:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (y <= -2.8e-25)
		tmp = Float64(x * y);
	elseif (y <= 9e+41)
		tmp = c;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (y <= -2.8e-25)
		tmp = x * y;
	elseif (y <= 9e+41)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[y, -2.8e-25], N[(x * y), $MachinePrecision], If[LessEqual[y, 9e+41], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.79999999999999988e-25 or 9.0000000000000002e41 < y

   1. Initial program 94.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 94.7%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 95.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 96.1%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 96.1%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]
   4. Step-by-step derivation

      1. fma-udef 95.3%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z}{16} \cdot t\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

      2. *-commutative 95.3%

         \[\leadsto \left(x \cdot y + \color{blue}{t \cdot \frac{z}{16}}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      3. div-inv 95.3%

         \[\leadsto \left(x \cdot y + t \cdot \color{blue}{\left(z \cdot \frac{1}{16}\right)}\right) - \left(\frac{a}{4} \cdot b - c\right) \]

      4. metadata-eval 95.3%

         \[\leadsto \left(x \cdot y + t \cdot \left(z \cdot \color{blue}{0.0625}\right)\right) - \left(\frac{a}{4} \cdot b - c\right) \]

   5. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\left(x \cdot y + t \cdot \left(z \cdot 0.0625\right)\right)} - \left(\frac{a}{4} \cdot b - c\right) \]

   6. Taylor expanded in x around inf 48.9%

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

   if -2.79999999999999988e-25 < y < 9.0000000000000002e41

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      2. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

      4. associate-*l/ 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

   4. Taylor expanded in c around inf 26.9%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-25}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+41}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 22.1% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. associate-+l- 97.3%

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

   2. fma-def 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   3. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t}\right) - \left(\frac{a \cdot b}{4} - c\right) \]

   4. associate-*l/ 98.0%

      \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{16} \cdot t\right) - \left(\frac{a}{4} \cdot b - c\right)} \]

4. Taylor expanded in c around inf 20.9%

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

5. Final simplification 20.9%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))