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

Percentage Accurate: 97.9% → 99.1%

Time: 8.1s

Alternatives: 12

Speedup: 1.0×

• 33.0% 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: 97.9% 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: 99.1% 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 96.9%

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

   1. associate-+l- 96.9%

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

   2. +-commutative 96.9%

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

   3. associate--l+ 96.9%

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

   4. associate-*l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. fma-def 99.2%

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

   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: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{t \cdot z}{16} + x \cdot y\right) - \frac{b \cdot a}{4} \leq \infty:\\ \;\;\;\;\left(z \cdot \left(t \cdot 0.0625\right) + x \cdot y\right) + \left(c - \frac{a}{\frac{4}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;c + \mathsf{fma}\left(y, x, b \cdot \left(a \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (- (+ (/ (* t z) 16.0) (* x y)) (/ (* b a) 4.0)) INFINITY)
   (+ (+ (* z (* t 0.0625)) (* x y)) (- c (/ a (/ 4.0 b))))
   (+ c (fma y x (* b (* a -0.25))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((((t * z) / 16.0) + (x * y)) - ((b * a) / 4.0)) <= ((double) INFINITY)) {
		tmp = ((z * (t * 0.0625)) + (x * y)) + (c - (a / (4.0 / b)));
	} else {
		tmp = c + fma(y, x, (b * (a * -0.25)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t * z) / 16.0) + Float64(x * y)) - Float64(Float64(b * a) / 4.0)) <= Inf)
		tmp = Float64(Float64(Float64(z * Float64(t * 0.0625)) + Float64(x * y)) + Float64(c - Float64(a / Float64(4.0 / b))));
	else
		tmp = Float64(c + fma(y, x, Float64(b * Float64(a * -0.25))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(c - N[(a / N[(4.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(y * x + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) 16)) (/.f64 (*.f64 a b) 4)) < +inf.0

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.9%

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

      6. clear-num 99.9%

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

      7. div-inv 99.9%

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

      8. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) 16)) (/.f64 (*.f64 a b) 4))

   1. Initial program 0.0%

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

   2. Taylor expanded in z around 0 71.4%

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

      1. fma-neg 71.4%

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

      2. distribute-lft-neg-in 71.4%

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

      3. metadata-eval 71.4%

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

      4. *-commutative 71.4%

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

      5. *-commutative 71.4%

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

      6. associate-*r* 71.4%

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

   4. Simplified 71.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, b \cdot \left(a \cdot -0.25\right)\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{t \cdot z}{16} + x \cdot y\right) - \frac{b \cdot a}{4} \leq \infty:\\ \;\;\;\;\left(z \cdot \left(t \cdot 0.0625\right) + x \cdot y\right) + \left(c - \frac{a}{\frac{4}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;c + \mathsf{fma}\left(y, x, b \cdot \left(a \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+158} \lor \neg \left(b \cdot a \leq 10^{+93}\right):\\ \;\;\;\;\left(c + t_1\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(t_1 + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (or (<= (* b a) -2e+158) (not (<= (* b a) 1e+93)))
     (- (+ c t_1) (* (* b a) 0.25))
     (+ c (+ 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 tmp;
	if (((b * a) <= -2e+158) || !((b * a) <= 1e+93)) {
		tmp = (c + t_1) - ((b * a) * 0.25);
	} else {
		tmp = c + (t_1 + (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) :: tmp
    t_1 = 0.0625d0 * (t * z)
    if (((b * a) <= (-2d+158)) .or. (.not. ((b * a) <= 1d+93))) then
        tmp = (c + t_1) - ((b * a) * 0.25d0)
    else
        tmp = c + (t_1 + (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 tmp;
	if (((b * a) <= -2e+158) || !((b * a) <= 1e+93)) {
		tmp = (c + t_1) - ((b * a) * 0.25);
	} else {
		tmp = c + (t_1 + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (t * z)
	tmp = 0
	if ((b * a) <= -2e+158) or not ((b * a) <= 1e+93):
		tmp = (c + t_1) - ((b * a) * 0.25)
	else:
		tmp = c + (t_1 + (x * y))
	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) <= -2e+158) || !(Float64(b * a) <= 1e+93))
		tmp = Float64(Float64(c + t_1) - Float64(Float64(b * a) * 0.25));
	else
		tmp = Float64(c + Float64(t_1 + 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);
	tmp = 0.0;
	if (((b * a) <= -2e+158) || ~(((b * a) <= 1e+93)))
		tmp = (c + t_1) - ((b * a) * 0.25);
	else
		tmp = c + (t_1 + (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]}, If[Or[LessEqual[N[(b * a), $MachinePrecision], -2e+158], N[Not[LessEqual[N[(b * a), $MachinePrecision], 1e+93]], $MachinePrecision]], N[(N[(c + t$95$1), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(c + N[(t$95$1 + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999991e158 or 1.00000000000000004e93 < (*.f64 a b)

   1. Initial program 94.6%

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

   2. Taylor expanded in x around 0 86.2%

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

   if -1.99999999999999991e158 < (*.f64 a b) < 1.00000000000000004e93

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0 90.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+158} \lor \neg \left(b \cdot a \leq 10^{+93}\right):\\ \;\;\;\;\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(0.0625 \cdot \left(t \cdot z\right) + x \cdot y\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+158} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+141}\right):\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(0.0625 \cdot \left(t \cdot z\right) + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* b a) -2e+158) (not (<= (* b a) 2e+141)))
   (+ c (* b (* a -0.25)))
   (+ c (+ (* 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 (((b * a) <= -2e+158) || !((b * a) <= 2e+141)) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = c + ((0.0625 * (t * z)) + (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 (((b * a) <= (-2d+158)) .or. (.not. ((b * a) <= 2d+141))) then
        tmp = c + (b * (a * (-0.25d0)))
    else
        tmp = c + ((0.0625d0 * (t * z)) + (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 (((b * a) <= -2e+158) || !((b * a) <= 2e+141)) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = c + ((0.0625 * (t * z)) + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((b * a) <= -2e+158) or not ((b * a) <= 2e+141):
		tmp = c + (b * (a * -0.25))
	else:
		tmp = c + ((0.0625 * (t * z)) + (x * y))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(b * a) <= -2e+158) || !(Float64(b * a) <= 2e+141))
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	else
		tmp = Float64(c + Float64(Float64(0.0625 * Float64(t * z)) + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((b * a) <= -2e+158) || ~(((b * a) <= 2e+141)))
		tmp = c + (b * (a * -0.25));
	else
		tmp = c + ((0.0625 * (t * z)) + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(b * a), $MachinePrecision], -2e+158], N[Not[LessEqual[N[(b * a), $MachinePrecision], 2e+141]], $MachinePrecision]], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999991e158 or 2.00000000000000003e141 < (*.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. Taylor expanded in a around inf 83.8%

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

      1. *-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*r* 83.8%

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

   4. Simplified 83.8%

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

   if -1.99999999999999991e158 < (*.f64 a b) < 2.00000000000000003e141

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 89.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+158} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+141}\right):\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(0.0625 \cdot \left(t \cdot z\right) + x \cdot y\right)\\ \end{array} \]

Alternative 5: 97.9% 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 96.9%

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

2. Final simplification 96.9%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((z * (t * 0.0625d0)) + (x * y)) + (c - (a / (4.0d0 / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. associate-+l- 96.9%

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

   2. sub-neg 96.9%

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

   3. neg-mul-1 96.9%

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

   4. metadata-eval 96.9%

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

   5. metadata-eval 96.9%

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

   6. cancel-sign-sub-inv 96.9%

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

   7. fma-def 97.3%

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

   8. associate-/l* 97.6%

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

   9. metadata-eval 97.6%

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

   10. *-lft-identity 97.6%

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

   11. associate-/l* 97.5%

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

3. Simplified 97.5%

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

   1. fma-udef 97.1%

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

   2. associate-/l* 96.9%

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

   3. +-commutative 96.9%

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

   4. associate-/l* 97.1%

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

   5. div-inv 97.2%

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

   6. clear-num 97.2%

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

   7. div-inv 97.2%

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

   8. metadata-eval 97.2%

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

5. Applied egg-rr 97.2%

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

6. Final simplification 97.2%

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

Alternative 7: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;b \leq -3.25 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-256}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 0.5:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= b -3.25e-118)
     t_1
     (if (<= b 1.12e-256)
       c
       (if (<= b 1.2e-109)
         (* x y)
         (if (<= b 0.5) c (if (<= b 8.5e+96) (* x y) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -3.25e-118) {
		tmp = t_1;
	} else if (b <= 1.12e-256) {
		tmp = c;
	} else if (b <= 1.2e-109) {
		tmp = x * y;
	} else if (b <= 0.5) {
		tmp = c;
	} else if (b <= 8.5e+96) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	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 = a * (b * (-0.25d0))
    if (b <= (-3.25d-118)) then
        tmp = t_1
    else if (b <= 1.12d-256) then
        tmp = c
    else if (b <= 1.2d-109) then
        tmp = x * y
    else if (b <= 0.5d0) then
        tmp = c
    else if (b <= 8.5d+96) then
        tmp = x * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -3.25e-118) {
		tmp = t_1;
	} else if (b <= 1.12e-256) {
		tmp = c;
	} else if (b <= 1.2e-109) {
		tmp = x * y;
	} else if (b <= 0.5) {
		tmp = c;
	} else if (b <= 8.5e+96) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = a * (b * -0.25)
	tmp = 0
	if b <= -3.25e-118:
		tmp = t_1
	elif b <= 1.12e-256:
		tmp = c
	elif b <= 1.2e-109:
		tmp = x * y
	elif b <= 0.5:
		tmp = c
	elif b <= 8.5e+96:
		tmp = x * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (b <= -3.25e-118)
		tmp = t_1;
	elseif (b <= 1.12e-256)
		tmp = c;
	elseif (b <= 1.2e-109)
		tmp = Float64(x * y);
	elseif (b <= 0.5)
		tmp = c;
	elseif (b <= 8.5e+96)
		tmp = Float64(x * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = a * (b * -0.25);
	tmp = 0.0;
	if (b <= -3.25e-118)
		tmp = t_1;
	elseif (b <= 1.12e-256)
		tmp = c;
	elseif (b <= 1.2e-109)
		tmp = x * y;
	elseif (b <= 0.5)
		tmp = c;
	elseif (b <= 8.5e+96)
		tmp = x * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.25e-118], t$95$1, If[LessEqual[b, 1.12e-256], c, If[LessEqual[b, 1.2e-109], N[(x * y), $MachinePrecision], If[LessEqual[b, 0.5], c, If[LessEqual[b, 8.5e+96], N[(x * y), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.24999999999999979e-118 or 8.50000000000000025e96 < b

   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. sub-neg 95.7%

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

      3. neg-mul-1 95.7%

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

      4. metadata-eval 95.7%

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

      5. metadata-eval 95.7%

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

      6. cancel-sign-sub-inv 95.7%

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

      7. fma-def 96.4%

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

      8. associate-/l* 96.3%

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

      9. metadata-eval 96.3%

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

      10. *-lft-identity 96.3%

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

      11. associate-/l* 96.2%

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

   3. Simplified 96.2%

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

      1. fma-udef 95.5%

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

      2. associate-/l* 95.6%

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

      3. +-commutative 95.6%

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

      4. associate-/l* 95.5%

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

      5. div-inv 95.5%

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

      6. clear-num 95.6%

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

      7. div-inv 95.6%

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

      8. metadata-eval 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in a around inf 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-*l* 49.3%

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

   8. Simplified 49.3%

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

   if -3.24999999999999979e-118 < b < 1.12e-256 or 1.19999999999999994e-109 < b < 0.5

   1. Initial program 97.1%

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

   2. Taylor expanded in c around inf 40.1%

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

   if 1.12e-256 < b < 1.19999999999999994e-109 or 0.5 < b < 8.50000000000000025e96

   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. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. fma-def 100.0%

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

      8. associate-/l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. *-lft-identity 99.8%

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

      11. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-/l* 99.8%

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

      5. div-inv 100.0%

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

      6. clear-num 100.0%

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

      7. div-inv 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 41.0%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-118}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-256}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 0.5:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \end{array} \]

Alternative 8: 59.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{if}\;b \leq -1.08 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-257}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+97}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* b (* a -0.25)))))
   (if (<= b -1.08e-35)
     t_1
     (if (<= b 1.3e-257)
       (+ c (* 0.0625 (* t z)))
       (if (<= b 5e+97) (+ c (* x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (b * (a * -0.25));
	double tmp;
	if (b <= -1.08e-35) {
		tmp = t_1;
	} else if (b <= 1.3e-257) {
		tmp = c + (0.0625 * (t * z));
	} else if (b <= 5e+97) {
		tmp = c + (x * y);
	} else {
		tmp = t_1;
	}
	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 = c + (b * (a * (-0.25d0)))
    if (b <= (-1.08d-35)) then
        tmp = t_1
    else if (b <= 1.3d-257) then
        tmp = c + (0.0625d0 * (t * z))
    else if (b <= 5d+97) then
        tmp = c + (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + (b * (a * -0.25));
	double tmp;
	if (b <= -1.08e-35) {
		tmp = t_1;
	} else if (b <= 1.3e-257) {
		tmp = c + (0.0625 * (t * z));
	} else if (b <= 5e+97) {
		tmp = c + (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (b * (a * -0.25))
	tmp = 0
	if b <= -1.08e-35:
		tmp = t_1
	elif b <= 1.3e-257:
		tmp = c + (0.0625 * (t * z))
	elif b <= 5e+97:
		tmp = c + (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(b * Float64(a * -0.25)))
	tmp = 0.0
	if (b <= -1.08e-35)
		tmp = t_1;
	elseif (b <= 1.3e-257)
		tmp = Float64(c + Float64(0.0625 * Float64(t * z)));
	elseif (b <= 5e+97)
		tmp = Float64(c + Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (b * (a * -0.25));
	tmp = 0.0;
	if (b <= -1.08e-35)
		tmp = t_1;
	elseif (b <= 1.3e-257)
		tmp = c + (0.0625 * (t * z));
	elseif (b <= 5e+97)
		tmp = c + (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.08e-35], t$95$1, If[LessEqual[b, 1.3e-257], N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+97], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.08000000000000003e-35 or 4.99999999999999999e97 < b

   1. Initial program 94.8%

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

   2. Taylor expanded in a around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   4. Simplified 65.4%

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

   if -1.08000000000000003e-35 < b < 1.3e-257

   1. Initial program 98.5%

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

   2. Taylor expanded in z around inf 66.7%

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

   if 1.3e-257 < b < 4.99999999999999999e97

   1. Initial program 98.8%

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

   2. Taylor expanded in x around inf 63.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.08 \cdot 10^{-35}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-257}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+97}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \end{array} \]

Alternative 9: 54.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-257}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{+122}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= b -4.5e+22)
     t_1
     (if (<= b 4.4e-257)
       (+ c (* 0.0625 (* t z)))
       (if (<= b 9.4e+122) (+ c (* x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -4.5e+22) {
		tmp = t_1;
	} else if (b <= 4.4e-257) {
		tmp = c + (0.0625 * (t * z));
	} else if (b <= 9.4e+122) {
		tmp = c + (x * y);
	} else {
		tmp = t_1;
	}
	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 = a * (b * (-0.25d0))
    if (b <= (-4.5d+22)) then
        tmp = t_1
    else if (b <= 4.4d-257) then
        tmp = c + (0.0625d0 * (t * z))
    else if (b <= 9.4d+122) then
        tmp = c + (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -4.5e+22) {
		tmp = t_1;
	} else if (b <= 4.4e-257) {
		tmp = c + (0.0625 * (t * z));
	} else if (b <= 9.4e+122) {
		tmp = c + (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = a * (b * -0.25)
	tmp = 0
	if b <= -4.5e+22:
		tmp = t_1
	elif b <= 4.4e-257:
		tmp = c + (0.0625 * (t * z))
	elif b <= 9.4e+122:
		tmp = c + (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (b <= -4.5e+22)
		tmp = t_1;
	elseif (b <= 4.4e-257)
		tmp = Float64(c + Float64(0.0625 * Float64(t * z)));
	elseif (b <= 9.4e+122)
		tmp = Float64(c + Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = a * (b * -0.25);
	tmp = 0.0;
	if (b <= -4.5e+22)
		tmp = t_1;
	elseif (b <= 4.4e-257)
		tmp = c + (0.0625 * (t * z));
	elseif (b <= 9.4e+122)
		tmp = c + (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.5e+22], t$95$1, If[LessEqual[b, 4.4e-257], N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.4e+122], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999998e22 or 9.40000000000000047e122 < b

   1. Initial program 94.8%

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

      1. associate-+l- 94.8%

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

      2. sub-neg 94.8%

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

      3. neg-mul-1 94.8%

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

      4. metadata-eval 94.8%

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

      5. metadata-eval 94.8%

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

      6. cancel-sign-sub-inv 94.8%

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

      7. fma-def 95.8%

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

      8. associate-/l* 95.8%

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

      9. metadata-eval 95.8%

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

      10. *-lft-identity 95.8%

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

      11. associate-/l* 95.6%

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

   3. Simplified 95.6%

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

      1. fma-udef 94.6%

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

      2. associate-/l* 94.6%

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

      3. +-commutative 94.6%

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

      4. associate-/l* 94.6%

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

      5. div-inv 94.6%

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

      6. clear-num 94.6%

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

      7. div-inv 94.6%

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

      8. metadata-eval 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in a around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 63.3%

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

   8. Simplified 63.3%

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

   if -4.4999999999999998e22 < b < 4.39999999999999975e-257

   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 z around inf 62.9%

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

   if 4.39999999999999975e-257 < b < 9.40000000000000047e122

   1. Initial program 97.7%

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

   2. Taylor expanded in x around inf 63.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-257}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{+122}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \end{array} \]

Alternative 10: 55.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+38} \lor \neg \left(b \leq 1.28 \cdot 10^{+123}\right):\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= b -2.5e+38) (not (<= b 1.28e+123)))
   (* a (* b -0.25))
   (+ c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((b <= -2.5e+38) || !(b <= 1.28e+123)) {
		tmp = a * (b * -0.25);
	} else {
		tmp = c + (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 ((b <= (-2.5d+38)) .or. (.not. (b <= 1.28d+123))) then
        tmp = a * (b * (-0.25d0))
    else
        tmp = c + (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 ((b <= -2.5e+38) || !(b <= 1.28e+123)) {
		tmp = a * (b * -0.25);
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (b <= -2.5e+38) or not (b <= 1.28e+123):
		tmp = a * (b * -0.25)
	else:
		tmp = c + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((b <= -2.5e+38) || !(b <= 1.28e+123))
		tmp = Float64(a * Float64(b * -0.25));
	else
		tmp = Float64(c + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((b <= -2.5e+38) || ~((b <= 1.28e+123)))
		tmp = a * (b * -0.25);
	else
		tmp = c + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[b, -2.5e+38], N[Not[LessEqual[b, 1.28e+123]], $MachinePrecision]], N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.49999999999999985e38 or 1.28000000000000005e123 < b

   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. sub-neg 94.7%

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

      3. neg-mul-1 94.7%

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

      4. metadata-eval 94.7%

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

      5. metadata-eval 94.7%

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

      6. cancel-sign-sub-inv 94.7%

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

      7. fma-def 95.8%

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

      8. associate-/l* 95.7%

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

      9. metadata-eval 95.7%

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

      10. *-lft-identity 95.7%

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

      11. associate-/l* 95.6%

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

   3. Simplified 95.6%

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

      1. fma-udef 94.5%

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

      2. associate-/l* 94.6%

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

      3. +-commutative 94.6%

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

      4. associate-/l* 94.5%

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

      5. div-inv 94.6%

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

      6. clear-num 94.6%

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

      7. div-inv 94.6%

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

      8. metadata-eval 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in a around inf 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 63.9%

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

   8. Simplified 63.9%

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

   if -2.49999999999999985e38 < b < 1.28000000000000005e123

   1. Initial program 98.2%

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

   2. Taylor expanded in x around inf 60.8%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+38} \lor \neg \left(b \leq 1.28 \cdot 10^{+123}\right):\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 11: 37.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+67}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -2.05e+67) c (if (<= c 8.5e+27) (* x y) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -2.05e+67) {
		tmp = c;
	} else if (c <= 8.5e+27) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	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 (c <= (-2.05d+67)) then
        tmp = c
    else if (c <= 8.5d+27) then
        tmp = x * y
    else
        tmp = c
    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 (c <= -2.05e+67) {
		tmp = c;
	} else if (c <= 8.5e+27) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if c <= -2.05e+67:
		tmp = c
	elif c <= 8.5e+27:
		tmp = x * y
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= -2.05e+67)
		tmp = c;
	elseif (c <= 8.5e+27)
		tmp = Float64(x * y);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (c <= -2.05e+67)
		tmp = c;
	elseif (c <= 8.5e+27)
		tmp = x * y;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, -2.05e+67], c, If[LessEqual[c, 8.5e+27], N[(x * y), $MachinePrecision], c]]

Derivation

1. Split input into 2 regimes

2. if c < -2.0499999999999999e67 or 8.5e27 < c

   1. Initial program 96.9%

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

   2. Taylor expanded in c around inf 49.0%

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

   if -2.0499999999999999e67 < c < 8.5e27

   1. Initial program 97.0%

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

      1. associate-+l- 97.0%

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

      2. sub-neg 97.0%

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

      3. neg-mul-1 97.0%

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

      4. metadata-eval 97.0%

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

      5. metadata-eval 97.0%

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

      6. cancel-sign-sub-inv 97.0%

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

      7. fma-def 97.0%

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

      8. associate-/l* 97.4%

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

      9. metadata-eval 97.4%

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

      10. *-lft-identity 97.4%

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

      11. associate-/l* 97.3%

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

   3. Simplified 97.3%

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

      1. fma-udef 97.3%

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

      2. associate-/l* 96.9%

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

      3. +-commutative 96.9%

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

      4. associate-/l* 97.3%

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

      5. div-inv 97.4%

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

      6. clear-num 97.4%

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

      7. div-inv 97.4%

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

      8. metadata-eval 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in x around inf 33.8%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+67}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 12: 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 96.9%

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

2. Taylor expanded in c around inf 22.5%

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

3. Final simplification 22.5%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023200 
(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))