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

Percentage Accurate: 98.0% → 98.9%

Time: 12.2s

Alternatives: 14

Speedup: 0.5×

• 32.0% 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: 98.0% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, c + \left(a \cdot b\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (/ (* z t) 16.0) (* x y)) (/ (* a b) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (fma x y (+ c (* (* a b) -0.25))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = fma(x, y, (c + ((a * b) * -0.25)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(z * t) / 16.0) + Float64(x * y)) - Float64(Float64(a * b) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = fma(x, y, Float64(c + Float64(Float64(a * b) * -0.25)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(x * y + N[(c + N[(N[(a * b), $MachinePrecision] * -0.25), $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 100.0%

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

   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. Step-by-step derivation

      1. associate-+l- 0.0%

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

      2. associate--l+ 0.0%

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

      3. fma-def 33.3%

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

      4. associate-*l/ 33.3%

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

      5. fma-neg 44.4%

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

      6. sub-neg 44.4%

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

      7. distribute-neg-in 44.4%

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

      8. remove-double-neg 44.4%

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

      9. associate-/l* 44.4%

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

      10. distribute-frac-neg 44.4%

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

      11. associate-/r/ 44.4%

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

      12. fma-def 44.4%

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

      13. neg-mul-1 44.4%

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

      14. *-commutative 44.4%

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

      15. associate-/l* 44.4%

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

      16. metadata-eval 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in z around 0 55.6%

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

4. Final simplification 98.4%

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

Alternative 2: 99.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. associate-+l- 96.5%

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

   2. associate--l+ 96.5%

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

   3. fma-def 97.7%

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

   4. associate-*l/ 97.7%

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

   5. fma-neg 98.0%

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

   6. sub-neg 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. remove-double-neg 98.0%

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

   9. associate-/l* 98.0%

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

   10. distribute-frac-neg 98.0%

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

   11. associate-/r/ 98.0%

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

   12. fma-def 98.0%

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

   13. neg-mul-1 98.0%

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

   14. *-commutative 98.0%

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

   15. associate-/l* 98.0%

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

   16. metadata-eval 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot \left(b \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (/ (* z t) 16.0) (* x y)) (/ (* a b) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (fma y x (* a (* b -0.25))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = fma(y, x, (a * (b * -0.25)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(z * t) / 16.0) + Float64(x * y)) - Float64(Float64(a * b) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = fma(y, x, Float64(a * Float64(b * -0.25)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(y * x + N[(a * N[(b * -0.25), $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 100.0%

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

   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. Step-by-step derivation

      1. associate-+l- 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/l* 0.0%

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

      5. associate-/l* 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in t around 0 22.2%

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

   5. Taylor expanded in c around 0 22.2%

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

      1. fma-neg 55.6%

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

      2. distribute-lft-neg-in 55.6%

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

      3. metadata-eval 55.6%

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

      4. *-commutative 55.6%

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

      5. associate-*l* 55.6%

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

   7. Simplified 55.6%

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

4. Final simplification 98.4%

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

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (/ (* z t) 16.0) (* x y)) (/ (* a b) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (* a (* b -0.25)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = a * (b * -0.25);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = c + t_1;
	} else {
		tmp = a * (b * -0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0)
	tmp = 0
	if t_1 <= math.inf:
		tmp = c + t_1
	else:
		tmp = a * (b * -0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(z * t) / 16.0) + Float64(x * y)) - Float64(Float64(a * b) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = Float64(a * Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = c + t_1;
	else
		tmp = a * (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(a * N[(b * -0.25), $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 100.0%

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

   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. Step-by-step derivation

      1. associate-+l- 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/l* 0.0%

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

      5. associate-/l* 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 33.3%

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

   5. Taylor expanded in a around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-*l* 44.6%

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

   7. Simplified 44.6%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 5: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + t \cdot \left(z \cdot 0.0625\right)\\ t_2 := c + \left(a \cdot b\right) \cdot -0.25\\ t_3 := c + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 10^{-99}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* t (* z 0.0625))))
        (t_2 (+ c (* (* a b) -0.25)))
        (t_3 (+ c (* x y))))
   (if (<= (* a b) -2e+56)
     t_2
     (if (<= (* a b) -5e-12)
       t_1
       (if (<= (* a b) -5e-112)
         t_3
         (if (<= (* a b) 0.0)
           t_1
           (if (<= (* a b) 1e-99) t_3 (if (<= (* a b) 1e+107) 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 + (t * (z * 0.0625));
	double t_2 = c + ((a * b) * -0.25);
	double t_3 = c + (x * y);
	double tmp;
	if ((a * b) <= -2e+56) {
		tmp = t_2;
	} else if ((a * b) <= -5e-12) {
		tmp = t_1;
	} else if ((a * b) <= -5e-112) {
		tmp = t_3;
	} else if ((a * b) <= 0.0) {
		tmp = t_1;
	} else if ((a * b) <= 1e-99) {
		tmp = t_3;
	} else if ((a * b) <= 1e+107) {
		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 + (t * (z * 0.0625d0))
    t_2 = c + ((a * b) * (-0.25d0))
    t_3 = c + (x * y)
    if ((a * b) <= (-2d+56)) then
        tmp = t_2
    else if ((a * b) <= (-5d-12)) then
        tmp = t_1
    else if ((a * b) <= (-5d-112)) then
        tmp = t_3
    else if ((a * b) <= 0.0d0) then
        tmp = t_1
    else if ((a * b) <= 1d-99) then
        tmp = t_3
    else if ((a * b) <= 1d+107) 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 + (t * (z * 0.0625));
	double t_2 = c + ((a * b) * -0.25);
	double t_3 = c + (x * y);
	double tmp;
	if ((a * b) <= -2e+56) {
		tmp = t_2;
	} else if ((a * b) <= -5e-12) {
		tmp = t_1;
	} else if ((a * b) <= -5e-112) {
		tmp = t_3;
	} else if ((a * b) <= 0.0) {
		tmp = t_1;
	} else if ((a * b) <= 1e-99) {
		tmp = t_3;
	} else if ((a * b) <= 1e+107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (t * (z * 0.0625))
	t_2 = c + ((a * b) * -0.25)
	t_3 = c + (x * y)
	tmp = 0
	if (a * b) <= -2e+56:
		tmp = t_2
	elif (a * b) <= -5e-12:
		tmp = t_1
	elif (a * b) <= -5e-112:
		tmp = t_3
	elif (a * b) <= 0.0:
		tmp = t_1
	elif (a * b) <= 1e-99:
		tmp = t_3
	elif (a * b) <= 1e+107:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(t * Float64(z * 0.0625)))
	t_2 = Float64(c + Float64(Float64(a * b) * -0.25))
	t_3 = Float64(c + Float64(x * y))
	tmp = 0.0
	if (Float64(a * b) <= -2e+56)
		tmp = t_2;
	elseif (Float64(a * b) <= -5e-12)
		tmp = t_1;
	elseif (Float64(a * b) <= -5e-112)
		tmp = t_3;
	elseif (Float64(a * b) <= 0.0)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e-99)
		tmp = t_3;
	elseif (Float64(a * b) <= 1e+107)
		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 + (t * (z * 0.0625));
	t_2 = c + ((a * b) * -0.25);
	t_3 = c + (x * y);
	tmp = 0.0;
	if ((a * b) <= -2e+56)
		tmp = t_2;
	elseif ((a * b) <= -5e-12)
		tmp = t_1;
	elseif ((a * b) <= -5e-112)
		tmp = t_3;
	elseif ((a * b) <= 0.0)
		tmp = t_1;
	elseif ((a * b) <= 1e-99)
		tmp = t_3;
	elseif ((a * b) <= 1e+107)
		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[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+56], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], -5e-12], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], -5e-112], t$95$3, If[LessEqual[N[(a * b), $MachinePrecision], 0.0], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e-99], t$95$3, If[LessEqual[N[(a * b), $MachinePrecision], 1e+107], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000018e56 or 9.9999999999999997e106 < (*.f64 a b)

   1. Initial program 91.4%

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

   2. Taylor expanded in a around inf 72.8%

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

      1. *-commutative 72.8%

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

   4. Simplified 72.8%

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

   if -2.00000000000000018e56 < (*.f64 a b) < -4.9999999999999997e-12 or -5.00000000000000044e-112 < (*.f64 a b) < 0.0 or 1e-99 < (*.f64 a b) < 9.9999999999999997e106

   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 74.4%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r* 74.4%

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

   4. Simplified 74.4%

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

   if -4.9999999999999997e-12 < (*.f64 a b) < -5.00000000000000044e-112 or 0.0 < (*.f64 a b) < 1e-99

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 84.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+56}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-12}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-112}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{-99}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 10^{+107}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \end{array} \]

Alternative 6: 37.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-51}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-102}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))) (t_2 (* a (* b -0.25))))
   (if (<= y -2.7e-51)
     (* x y)
     (if (<= y -1.5e-295)
       t_1
       (if (<= y 2.2e-245)
         t_2
         (if (<= y 4.2e-205)
           t_1
           (if (<= y 3.8e-102)
             c
             (if (<= y 2.6e-83)
               t_1
               (if (<= y 2e+14) c (if (<= y 1e+138) t_2 (* x y)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (z * t);
	double t_2 = a * (b * -0.25);
	double tmp;
	if (y <= -2.7e-51) {
		tmp = x * y;
	} else if (y <= -1.5e-295) {
		tmp = t_1;
	} else if (y <= 2.2e-245) {
		tmp = t_2;
	} else if (y <= 4.2e-205) {
		tmp = t_1;
	} else if (y <= 3.8e-102) {
		tmp = c;
	} else if (y <= 2.6e-83) {
		tmp = t_1;
	} else if (y <= 2e+14) {
		tmp = c;
	} else if (y <= 1e+138) {
		tmp = t_2;
	} 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 * (z * t)
    t_2 = a * (b * (-0.25d0))
    if (y <= (-2.7d-51)) then
        tmp = x * y
    else if (y <= (-1.5d-295)) then
        tmp = t_1
    else if (y <= 2.2d-245) then
        tmp = t_2
    else if (y <= 4.2d-205) then
        tmp = t_1
    else if (y <= 3.8d-102) then
        tmp = c
    else if (y <= 2.6d-83) then
        tmp = t_1
    else if (y <= 2d+14) then
        tmp = c
    else if (y <= 1d+138) then
        tmp = t_2
    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 * (z * t);
	double t_2 = a * (b * -0.25);
	double tmp;
	if (y <= -2.7e-51) {
		tmp = x * y;
	} else if (y <= -1.5e-295) {
		tmp = t_1;
	} else if (y <= 2.2e-245) {
		tmp = t_2;
	} else if (y <= 4.2e-205) {
		tmp = t_1;
	} else if (y <= 3.8e-102) {
		tmp = c;
	} else if (y <= 2.6e-83) {
		tmp = t_1;
	} else if (y <= 2e+14) {
		tmp = c;
	} else if (y <= 1e+138) {
		tmp = t_2;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (z * t)
	t_2 = a * (b * -0.25)
	tmp = 0
	if y <= -2.7e-51:
		tmp = x * y
	elif y <= -1.5e-295:
		tmp = t_1
	elif y <= 2.2e-245:
		tmp = t_2
	elif y <= 4.2e-205:
		tmp = t_1
	elif y <= 3.8e-102:
		tmp = c
	elif y <= 2.6e-83:
		tmp = t_1
	elif y <= 2e+14:
		tmp = c
	elif y <= 1e+138:
		tmp = t_2
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(z * t))
	t_2 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (y <= -2.7e-51)
		tmp = Float64(x * y);
	elseif (y <= -1.5e-295)
		tmp = t_1;
	elseif (y <= 2.2e-245)
		tmp = t_2;
	elseif (y <= 4.2e-205)
		tmp = t_1;
	elseif (y <= 3.8e-102)
		tmp = c;
	elseif (y <= 2.6e-83)
		tmp = t_1;
	elseif (y <= 2e+14)
		tmp = c;
	elseif (y <= 1e+138)
		tmp = t_2;
	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 * (z * t);
	t_2 = a * (b * -0.25);
	tmp = 0.0;
	if (y <= -2.7e-51)
		tmp = x * y;
	elseif (y <= -1.5e-295)
		tmp = t_1;
	elseif (y <= 2.2e-245)
		tmp = t_2;
	elseif (y <= 4.2e-205)
		tmp = t_1;
	elseif (y <= 3.8e-102)
		tmp = c;
	elseif (y <= 2.6e-83)
		tmp = t_1;
	elseif (y <= 2e+14)
		tmp = c;
	elseif (y <= 1e+138)
		tmp = t_2;
	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[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e-51], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.5e-295], t$95$1, If[LessEqual[y, 2.2e-245], t$95$2, If[LessEqual[y, 4.2e-205], t$95$1, If[LessEqual[y, 3.8e-102], c, If[LessEqual[y, 2.6e-83], t$95$1, If[LessEqual[y, 2e+14], c, If[LessEqual[y, 1e+138], t$95$2, N[(x * y), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.6999999999999997e-51 or 1e138 < y

   1. Initial program 92.2%

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

   2. Taylor expanded in x around inf 59.0%

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

   3. Taylor expanded in y around inf 44.1%

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

   if -2.6999999999999997e-51 < y < -1.49999999999999998e-295 or 2.19999999999999993e-245 < y < 4.19999999999999965e-205 or 3.80000000000000026e-102 < y < 2.60000000000000009e-83

   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. *-commutative 98.7%

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

      4. associate-/l* 98.6%

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

      5. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around 0 94.0%

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

   5. Taylor expanded in t around inf 38.6%

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

   if -1.49999999999999998e-295 < y < 2.19999999999999993e-245 or 2e14 < y < 1e138

   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. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 79.0%

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

   5. Taylor expanded in a around inf 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-*l* 36.4%

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

   7. Simplified 36.4%

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

   if 4.19999999999999965e-205 < y < 3.80000000000000026e-102 or 2.60000000000000009e-83 < y < 2e14

   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. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in c around inf 47.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-51}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-295}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-245}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-205}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-102}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 10^{+138}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;c \leq -2.5 \cdot 10^{+36}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (* 0.0625 (* z t)) (* (* a b) 0.25))))
   (if (<= c -2.5e+36)
     (+ c (* t (* z 0.0625)))
     (if (<= c 1.95e-172)
       t_1
       (if (<= c 8.5e-143) (* x y) (if (<= c 1.55e+45) t_1 (+ c (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (0.0625 * (z * t)) - ((a * b) * 0.25);
	double tmp;
	if (c <= -2.5e+36) {
		tmp = c + (t * (z * 0.0625));
	} else if (c <= 1.95e-172) {
		tmp = t_1;
	} else if (c <= 8.5e-143) {
		tmp = x * y;
	} else if (c <= 1.55e+45) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (0.0625d0 * (z * t)) - ((a * b) * 0.25d0)
    if (c <= (-2.5d+36)) then
        tmp = c + (t * (z * 0.0625d0))
    else if (c <= 1.95d-172) then
        tmp = t_1
    else if (c <= 8.5d-143) then
        tmp = x * y
    else if (c <= 1.55d+45) then
        tmp = t_1
    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 t_1 = (0.0625 * (z * t)) - ((a * b) * 0.25);
	double tmp;
	if (c <= -2.5e+36) {
		tmp = c + (t * (z * 0.0625));
	} else if (c <= 1.95e-172) {
		tmp = t_1;
	} else if (c <= 8.5e-143) {
		tmp = x * y;
	} else if (c <= 1.55e+45) {
		tmp = t_1;
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (0.0625 * (z * t)) - ((a * b) * 0.25)
	tmp = 0
	if c <= -2.5e+36:
		tmp = c + (t * (z * 0.0625))
	elif c <= 1.95e-172:
		tmp = t_1
	elif c <= 8.5e-143:
		tmp = x * y
	elif c <= 1.55e+45:
		tmp = t_1
	else:
		tmp = c + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.0625 * Float64(z * t)) - Float64(Float64(a * b) * 0.25))
	tmp = 0.0
	if (c <= -2.5e+36)
		tmp = Float64(c + Float64(t * Float64(z * 0.0625)));
	elseif (c <= 1.95e-172)
		tmp = t_1;
	elseif (c <= 8.5e-143)
		tmp = Float64(x * y);
	elseif (c <= 1.55e+45)
		tmp = t_1;
	else
		tmp = Float64(c + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (0.0625 * (z * t)) - ((a * b) * 0.25);
	tmp = 0.0;
	if (c <= -2.5e+36)
		tmp = c + (t * (z * 0.0625));
	elseif (c <= 1.95e-172)
		tmp = t_1;
	elseif (c <= 8.5e-143)
		tmp = x * y;
	elseif (c <= 1.55e+45)
		tmp = t_1;
	else
		tmp = c + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.5e+36], N[(c + N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.95e-172], t$95$1, If[LessEqual[c, 8.5e-143], N[(x * y), $MachinePrecision], If[LessEqual[c, 1.55e+45], t$95$1, N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.49999999999999988e36

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. associate-*r* 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-*r* 74.5%

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

   4. Simplified 74.5%

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

   if -2.49999999999999988e36 < c < 1.94999999999999986e-172 or 8.50000000000000072e-143 < c < 1.54999999999999994e45

   1. Initial program 97.7%

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

      1. associate-+l- 97.7%

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

      2. fma-def 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. *-commutative 97.7%

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

      4. associate-/l* 97.6%

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

      5. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in c around 0 73.9%

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

   if 1.94999999999999986e-172 < c < 8.50000000000000072e-143

   1. Initial program 87.5%

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

   2. Taylor expanded in x around inf 87.6%

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

   3. Taylor expanded in y around inf 87.6%

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

   if 1.54999999999999994e45 < c

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 71.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{+36}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+45}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 8: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{-66} \lor \neg \left(z \leq 3.2 \cdot 10^{-116}\right):\\ \;\;\;\;\left(c + 0.0625 \cdot \left(z \cdot t\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)))
   (if (or (<= z -1.25e-66) (not (<= z 3.2e-116)))
     (- (+ c (* 0.0625 (* z t))) t_1)
     (- (+ 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 ((z <= -1.25e-66) || !(z <= 3.2e-116)) {
		tmp = (c + (0.0625 * (z * t))) - t_1;
	} else {
		tmp = (c + (x * y)) - 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 ((z <= (-1.25d-66)) .or. (.not. (z <= 3.2d-116))) then
        tmp = (c + (0.0625d0 * (z * t))) - t_1
    else
        tmp = (c + (x * y)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) * 0.25;
	double tmp;
	if ((z <= -1.25e-66) || !(z <= 3.2e-116)) {
		tmp = (c + (0.0625 * (z * t))) - t_1;
	} else {
		tmp = (c + (x * y)) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	tmp = 0
	if (z <= -1.25e-66) or not (z <= 3.2e-116):
		tmp = (c + (0.0625 * (z * t))) - t_1
	else:
		tmp = (c + (x * y)) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) * 0.25)
	tmp = 0.0
	if ((z <= -1.25e-66) || !(z <= 3.2e-116))
		tmp = Float64(Float64(c + Float64(0.0625 * Float64(z * t))) - t_1);
	else
		tmp = Float64(Float64(c + Float64(x * y)) - 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 ((z <= -1.25e-66) || ~((z <= 3.2e-116)))
		tmp = (c + (0.0625 * (z * t))) - t_1;
	else
		tmp = (c + (x * y)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]}, If[Or[LessEqual[z, -1.25e-66], N[Not[LessEqual[z, 3.2e-116]], $MachinePrecision]], N[(N[(c + N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2499999999999999e-66 or 3.20000000000000009e-116 < z

   1. Initial program 96.4%

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

      1. associate-+l- 96.4%

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

      2. fma-def 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-/l* 96.3%

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

      5. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in x around 0 82.3%

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

   if -1.2499999999999999e-66 < z < 3.20000000000000009e-116

   1. Initial program 96.6%

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

      1. associate-+l- 96.6%

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

      2. fma-def 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-/l* 96.6%

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

      5. associate-/l* 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in t around 0 90.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-66} \lor \neg \left(z \leq 3.2 \cdot 10^{-116}\right):\\ \;\;\;\;\left(c + 0.0625 \cdot \left(z \cdot t\right)\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 9: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-51} \lor \neg \left(y \leq 1.26 \cdot 10^{+116}\right) \land \left(y \leq 5.8 \cdot 10^{+149} \lor \neg \left(y \leq 2.25 \cdot 10^{+203}\right)\right):\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= y -1.1e-51)
         (and (not (<= y 1.26e+116))
              (or (<= y 5.8e+149) (not (<= y 2.25e+203)))))
   (+ c (* x y))
   (+ c (* t (* z 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((y <= -1.1e-51) || (!(y <= 1.26e+116) && ((y <= 5.8e+149) || !(y <= 2.25e+203)))) {
		tmp = c + (x * y);
	} else {
		tmp = c + (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) :: tmp
    if ((y <= (-1.1d-51)) .or. (.not. (y <= 1.26d+116)) .and. (y <= 5.8d+149) .or. (.not. (y <= 2.25d+203))) then
        tmp = c + (x * y)
    else
        tmp = c + (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 tmp;
	if ((y <= -1.1e-51) || (!(y <= 1.26e+116) && ((y <= 5.8e+149) || !(y <= 2.25e+203)))) {
		tmp = c + (x * y);
	} else {
		tmp = c + (t * (z * 0.0625));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (y <= -1.1e-51) or (not (y <= 1.26e+116) and ((y <= 5.8e+149) or not (y <= 2.25e+203))):
		tmp = c + (x * y)
	else:
		tmp = c + (t * (z * 0.0625))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((y <= -1.1e-51) || (!(y <= 1.26e+116) && ((y <= 5.8e+149) || !(y <= 2.25e+203))))
		tmp = Float64(c + Float64(x * y));
	else
		tmp = Float64(c + Float64(t * Float64(z * 0.0625)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((y <= -1.1e-51) || (~((y <= 1.26e+116)) && ((y <= 5.8e+149) || ~((y <= 2.25e+203)))))
		tmp = c + (x * y);
	else
		tmp = c + (t * (z * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[y, -1.1e-51], And[N[Not[LessEqual[y, 1.26e+116]], $MachinePrecision], Or[LessEqual[y, 5.8e+149], N[Not[LessEqual[y, 2.25e+203]], $MachinePrecision]]]], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(c + N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e-51 or 1.2599999999999999e116 < y < 5.8000000000000004e149 or 2.2500000000000001e203 < y

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 60.8%

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

   if -1.1e-51 < y < 1.2599999999999999e116 or 5.8000000000000004e149 < y < 2.2500000000000001e203

   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 z around inf 63.3%

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

      1. associate-*r* 63.3%

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

      2. *-commutative 63.3%

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

      3. associate-*r* 63.3%

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

   4. Simplified 63.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-51} \lor \neg \left(y \leq 1.26 \cdot 10^{+116}\right) \land \left(y \leq 5.8 \cdot 10^{+149} \lor \neg \left(y \leq 2.25 \cdot 10^{+203}\right)\right):\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \]

Alternative 10: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;t \leq -6.7 \cdot 10^{-50}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - t_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+135}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)))
   (if (<= t -6.7e-50)
     (- (* 0.0625 (* z t)) t_1)
     (if (<= t 2.65e+135) (- (+ c (* x y)) t_1) (+ c (* t (* z 0.0625)))))))

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 (t <= -6.7e-50) {
		tmp = (0.0625 * (z * t)) - t_1;
	} else if (t <= 2.65e+135) {
		tmp = (c + (x * y)) - t_1;
	} else {
		tmp = c + (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) :: tmp
    t_1 = (a * b) * 0.25d0
    if (t <= (-6.7d-50)) then
        tmp = (0.0625d0 * (z * t)) - t_1
    else if (t <= 2.65d+135) then
        tmp = (c + (x * y)) - t_1
    else
        tmp = c + (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 = (a * b) * 0.25;
	double tmp;
	if (t <= -6.7e-50) {
		tmp = (0.0625 * (z * t)) - t_1;
	} else if (t <= 2.65e+135) {
		tmp = (c + (x * y)) - t_1;
	} else {
		tmp = c + (t * (z * 0.0625));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	tmp = 0
	if t <= -6.7e-50:
		tmp = (0.0625 * (z * t)) - t_1
	elif t <= 2.65e+135:
		tmp = (c + (x * y)) - t_1
	else:
		tmp = c + (t * (z * 0.0625))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) * 0.25)
	tmp = 0.0
	if (t <= -6.7e-50)
		tmp = Float64(Float64(0.0625 * Float64(z * t)) - t_1);
	elseif (t <= 2.65e+135)
		tmp = Float64(Float64(c + Float64(x * y)) - t_1);
	else
		tmp = Float64(c + Float64(t * Float64(z * 0.0625)));
	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 (t <= -6.7e-50)
		tmp = (0.0625 * (z * t)) - t_1;
	elseif (t <= 2.65e+135)
		tmp = (c + (x * y)) - t_1;
	else
		tmp = c + (t * (z * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[t, -6.7e-50], N[(N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t, 2.65e+135], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(c + N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.7000000000000005e-50

   1. Initial program 94.4%

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

      1. associate-+l- 94.4%

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

      2. fma-def 94.4%

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

      3. *-commutative 94.4%

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

      4. associate-/l* 94.3%

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

      5. associate-/l* 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around 0 82.5%

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

   5. Taylor expanded in c around 0 66.4%

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

   if -6.7000000000000005e-50 < t < 2.65000000000000008e135

   1. Initial program 97.1%

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

      1. associate-+l- 97.1%

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

      2. fma-def 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-/l* 97.1%

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

      5. associate-/l* 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in t around 0 90.3%

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

   if 2.65000000000000008e135 < t

   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 83.0%

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

      1. associate-*r* 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*r* 83.0%

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

   4. Simplified 83.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{-50}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+135}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \]

Alternative 11: 38.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+74}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))))
   (if (<= y -3.9e-53)
     (* x y)
     (if (<= y 3.9e-203)
       t_1
       (if (<= y 1.4e-103)
         c
         (if (<= y 8e-83) t_1 (if (<= y 3.2e+74) c (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (z * t);
	double tmp;
	if (y <= -3.9e-53) {
		tmp = x * y;
	} else if (y <= 3.9e-203) {
		tmp = t_1;
	} else if (y <= 1.4e-103) {
		tmp = c;
	} else if (y <= 8e-83) {
		tmp = t_1;
	} else if (y <= 3.2e+74) {
		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) :: t_1
    real(8) :: tmp
    t_1 = 0.0625d0 * (z * t)
    if (y <= (-3.9d-53)) then
        tmp = x * y
    else if (y <= 3.9d-203) then
        tmp = t_1
    else if (y <= 1.4d-103) then
        tmp = c
    else if (y <= 8d-83) then
        tmp = t_1
    else if (y <= 3.2d+74) 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 t_1 = 0.0625 * (z * t);
	double tmp;
	if (y <= -3.9e-53) {
		tmp = x * y;
	} else if (y <= 3.9e-203) {
		tmp = t_1;
	} else if (y <= 1.4e-103) {
		tmp = c;
	} else if (y <= 8e-83) {
		tmp = t_1;
	} else if (y <= 3.2e+74) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (z * t)
	tmp = 0
	if y <= -3.9e-53:
		tmp = x * y
	elif y <= 3.9e-203:
		tmp = t_1
	elif y <= 1.4e-103:
		tmp = c
	elif y <= 8e-83:
		tmp = t_1
	elif y <= 3.2e+74:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (y <= -3.9e-53)
		tmp = Float64(x * y);
	elseif (y <= 3.9e-203)
		tmp = t_1;
	elseif (y <= 1.4e-103)
		tmp = c;
	elseif (y <= 8e-83)
		tmp = t_1;
	elseif (y <= 3.2e+74)
		tmp = c;
	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 * (z * t);
	tmp = 0.0;
	if (y <= -3.9e-53)
		tmp = x * y;
	elseif (y <= 3.9e-203)
		tmp = t_1;
	elseif (y <= 1.4e-103)
		tmp = c;
	elseif (y <= 8e-83)
		tmp = t_1;
	elseif (y <= 3.2e+74)
		tmp = c;
	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[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.9e-53], N[(x * y), $MachinePrecision], If[LessEqual[y, 3.9e-203], t$95$1, If[LessEqual[y, 1.4e-103], c, If[LessEqual[y, 8e-83], t$95$1, If[LessEqual[y, 3.2e+74], c, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9000000000000002e-53 or 3.19999999999999995e74 < y

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 56.9%

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

   3. Taylor expanded in y around inf 42.2%

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

   if -3.9000000000000002e-53 < y < 3.8999999999999999e-203 or 1.40000000000000011e-103 < y < 8.0000000000000003e-83

   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. *-commutative 98.9%

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

      4. associate-/l* 98.8%

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

      5. associate-/l* 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in x around 0 93.7%

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

   5. Taylor expanded in t around inf 35.2%

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

   if 3.8999999999999999e-203 < y < 1.40000000000000011e-103 or 8.0000000000000003e-83 < y < 3.19999999999999995e74

   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. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in c around inf 41.5%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-203}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-83}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+74}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 54.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;b \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+236} \lor \neg \left(b \leq 1.4 \cdot 10^{+252}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= b -9.2e+49)
     t_1
     (if (<= b 3.8e+153)
       (+ c (* x y))
       (if (or (<= b 3.8e+236) (not (<= b 1.4e+252)))
         t_1
         (* 0.0625 (* z t)))))))

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 <= -9.2e+49) {
		tmp = t_1;
	} else if (b <= 3.8e+153) {
		tmp = c + (x * y);
	} else if ((b <= 3.8e+236) || !(b <= 1.4e+252)) {
		tmp = t_1;
	} else {
		tmp = 0.0625 * (z * t);
	}
	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 <= (-9.2d+49)) then
        tmp = t_1
    else if (b <= 3.8d+153) then
        tmp = c + (x * y)
    else if ((b <= 3.8d+236) .or. (.not. (b <= 1.4d+252))) then
        tmp = t_1
    else
        tmp = 0.0625d0 * (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -9.2e+49) {
		tmp = t_1;
	} else if (b <= 3.8e+153) {
		tmp = c + (x * y);
	} else if ((b <= 3.8e+236) || !(b <= 1.4e+252)) {
		tmp = t_1;
	} else {
		tmp = 0.0625 * (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = a * (b * -0.25)
	tmp = 0
	if b <= -9.2e+49:
		tmp = t_1
	elif b <= 3.8e+153:
		tmp = c + (x * y)
	elif (b <= 3.8e+236) or not (b <= 1.4e+252):
		tmp = t_1
	else:
		tmp = 0.0625 * (z * t)
	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 <= -9.2e+49)
		tmp = t_1;
	elseif (b <= 3.8e+153)
		tmp = Float64(c + Float64(x * y));
	elseif ((b <= 3.8e+236) || !(b <= 1.4e+252))
		tmp = t_1;
	else
		tmp = Float64(0.0625 * Float64(z * t));
	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 <= -9.2e+49)
		tmp = t_1;
	elseif (b <= 3.8e+153)
		tmp = c + (x * y);
	elseif ((b <= 3.8e+236) || ~((b <= 1.4e+252)))
		tmp = t_1;
	else
		tmp = 0.0625 * (z * t);
	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, -9.2e+49], t$95$1, If[LessEqual[b, 3.8e+153], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 3.8e+236], N[Not[LessEqual[b, 1.4e+252]], $MachinePrecision]], t$95$1, N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.20000000000000008e49 or 3.79999999999999966e153 < b < 3.79999999999999986e236 or 1.40000000000000002e252 < b

   1. Initial program 91.6%

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

      1. associate-+l- 91.6%

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

      2. fma-def 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-/l* 91.5%

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

      5. associate-/l* 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around 0 86.0%

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

   5. Taylor expanded in a around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

   7. Simplified 52.5%

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

   if -9.20000000000000008e49 < b < 3.79999999999999966e153

   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 59.0%

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

   if 3.79999999999999986e236 < b < 1.40000000000000002e252

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. fma-def 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.2%

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

   5. Taylor expanded in t around inf 76.1%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+236} \lor \neg \left(b \leq 1.4 \cdot 10^{+252}\right):\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]

Alternative 13: 36.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= y -6e-90) (* x y) (if (<= y 1.45e+74) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (y <= -6e-90) {
		tmp = x * y;
	} else if (y <= 1.45e+74) {
		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 <= (-6d-90)) then
        tmp = x * y
    else if (y <= 1.45d+74) 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 <= -6e-90) {
		tmp = x * y;
	} else if (y <= 1.45e+74) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if y <= -6e-90:
		tmp = x * y
	elif y <= 1.45e+74:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (y <= -6e-90)
		tmp = Float64(x * y);
	elseif (y <= 1.45e+74)
		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 <= -6e-90)
		tmp = x * y;
	elseif (y <= 1.45e+74)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[y, -6e-90], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.45e+74], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000041e-90 or 1.4500000000000001e74 < y

   1. Initial program 93.7%

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

   2. Taylor expanded in x around inf 54.3%

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

   3. Taylor expanded in y around inf 39.1%

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

   if -6.00000000000000041e-90 < y < 1.4500000000000001e74

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. fma-def 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-/l* 99.2%

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

      5. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in c around inf 36.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14: 22.6% 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 96.5%

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

   1. associate-+l- 96.5%

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

   2. fma-def 96.5%

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

   3. *-commutative 96.5%

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

   4. associate-/l* 96.4%

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

   5. associate-/l* 96.4%

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

3. Simplified 96.4%

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

4. Taylor expanded in c around inf 26.6%

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

5. Final simplification 26.6%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023274 
(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))