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

Percentage Accurate: 97.5% → 98.9%

Time: 10.5s

Alternatives: 16

Speedup: 0.5×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $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] + N[(x * y + c), $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 20.0%

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

      3. *-commutative 20.0%

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

      4. associate-/l* 20.0%

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

      5. associate-/l* 20.0%

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

   3. Simplified 20.0%

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

      1. fma-udef 0.0%

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

      2. div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. div-inv 0.0%

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

      5. metadata-eval 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in t around 0 30.0%

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

      1. cancel-sign-sub-inv 30.0%

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

      2. +-commutative 30.0%

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

      3. fma-udef 30.0%

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

      4. metadata-eval 30.0%

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

      5. *-commutative 30.0%

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

      6. associate-*r* 30.0%

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

      7. +-commutative 30.0%

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

      8. fma-def 60.0%

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

   8. Simplified 60.0%

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

4. Final simplification 98.4%

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

Alternative 2: 98.8% 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 96.1%

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

   1. associate-+l- 96.1%

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

   2. associate--l+ 96.1%

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

   3. fma-def 97.2%

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

   4. associate-*l/ 97.2%

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

   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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;c + \mathsf{fma}\left(z, t \cdot 0.0625, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (+ c (fma z (* t 0.0625) (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = c + fma(z, (t * 0.0625), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = Float64(c + fma(z, Float64(t * 0.0625), Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(c + N[(z * N[(t * 0.0625), $MachinePrecision] + N[(x * y), $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. Taylor expanded in a around 0 40.0%

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

      1. *-commutative 40.0%

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

      2. *-commutative 40.0%

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

      3. associate-*l* 40.0%

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

      4. fma-def 50.0%

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

      5. *-commutative 50.0%

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

   4. Simplified 50.0%

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

4. Final simplification 98.0%

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

Alternative 4: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. associate-+l- 96.1%

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

   2. fma-def 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-/l* 96.8%

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

   5. associate-/l* 96.8%

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

3. Simplified 96.8%

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

4. Final simplification 96.8%

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

Alternative 5: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0))))
   (if (<= t_1 INFINITY) (+ 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 = ((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $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), $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 20.0%

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

      3. *-commutative 20.0%

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

      4. associate-/l* 20.0%

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

      5. associate-/l* 20.0%

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

   3. Simplified 20.0%

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

      1. fma-udef 0.0%

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

      2. div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. div-inv 0.0%

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

      5. metadata-eval 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x around inf 40.7%

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

4. Final simplification 97.7%

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

Alternative 6: 66.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + t \cdot \left(z \cdot 0.0625\right)\\ t_2 := x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;x \cdot y \leq -5.8 \cdot 10^{+163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{+46}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;x \cdot y \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* t (* z 0.0625)))) (t_2 (- (* x y) (* (* a b) 0.25))))
   (if (<= (* x y) -5.8e+163)
     t_2
     (if (<= (* x y) 1.2e-89)
       t_1
       (if (<= (* x y) 5.5e+46)
         (+ c (* (* a b) -0.25))
         (if (<= (* x y) 1.75e+102) 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 = (x * y) - ((a * b) * 0.25);
	double tmp;
	if ((x * y) <= -5.8e+163) {
		tmp = t_2;
	} else if ((x * y) <= 1.2e-89) {
		tmp = t_1;
	} else if ((x * y) <= 5.5e+46) {
		tmp = c + ((a * b) * -0.25);
	} else if ((x * y) <= 1.75e+102) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = c + (t * (z * 0.0625d0))
    t_2 = (x * y) - ((a * b) * 0.25d0)
    if ((x * y) <= (-5.8d+163)) then
        tmp = t_2
    else if ((x * y) <= 1.2d-89) then
        tmp = t_1
    else if ((x * y) <= 5.5d+46) then
        tmp = c + ((a * b) * (-0.25d0))
    else if ((x * y) <= 1.75d+102) 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 = (x * y) - ((a * b) * 0.25);
	double tmp;
	if ((x * y) <= -5.8e+163) {
		tmp = t_2;
	} else if ((x * y) <= 1.2e-89) {
		tmp = t_1;
	} else if ((x * y) <= 5.5e+46) {
		tmp = c + ((a * b) * -0.25);
	} else if ((x * y) <= 1.75e+102) {
		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 = (x * y) - ((a * b) * 0.25)
	tmp = 0
	if (x * y) <= -5.8e+163:
		tmp = t_2
	elif (x * y) <= 1.2e-89:
		tmp = t_1
	elif (x * y) <= 5.5e+46:
		tmp = c + ((a * b) * -0.25)
	elif (x * y) <= 1.75e+102:
		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(Float64(x * y) - Float64(Float64(a * b) * 0.25))
	tmp = 0.0
	if (Float64(x * y) <= -5.8e+163)
		tmp = t_2;
	elseif (Float64(x * y) <= 1.2e-89)
		tmp = t_1;
	elseif (Float64(x * y) <= 5.5e+46)
		tmp = Float64(c + Float64(Float64(a * b) * -0.25));
	elseif (Float64(x * y) <= 1.75e+102)
		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 = (x * y) - ((a * b) * 0.25);
	tmp = 0.0;
	if ((x * y) <= -5.8e+163)
		tmp = t_2;
	elseif ((x * y) <= 1.2e-89)
		tmp = t_1;
	elseif ((x * y) <= 5.5e+46)
		tmp = c + ((a * b) * -0.25);
	elseif ((x * y) <= 1.75e+102)
		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[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5.8e+163], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 1.2e-89], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5.5e+46], N[(c + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.75e+102], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.79999999999999996e163 or 1.75000000000000005e102 < (*.f64 x y)

   1. Initial program 90.5%

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

   2. Taylor expanded in z around 0 82.9%

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

   3. Taylor expanded in c around 0 80.3%

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

   if -5.79999999999999996e163 < (*.f64 x y) < 1.20000000000000008e-89 or 5.4999999999999998e46 < (*.f64 x y) < 1.75000000000000005e102

   1. Initial program 98.0%

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

   2. Taylor expanded in z around inf 72.2%

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

      1. associate-*r* 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-*r* 72.2%

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

   4. Simplified 72.2%

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

   if 1.20000000000000008e-89 < (*.f64 x y) < 5.4999999999999998e46

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 70.1%

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

      1. *-commutative 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.8 \cdot 10^{+163}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{-89}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{+46}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;x \cdot y \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 7: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+168}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.6 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;x \cdot y \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* t (* z 0.0625)))))
   (if (<= (* x y) -2.2e+168)
     (* x y)
     (if (<= (* x y) 7.6e-91)
       t_1
       (if (<= (* x y) 1.2e+49)
         (+ c (* (* a b) -0.25))
         (if (<= (* x y) 2.15e+102) 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 = c + (t * (z * 0.0625));
	double tmp;
	if ((x * y) <= -2.2e+168) {
		tmp = x * y;
	} else if ((x * y) <= 7.6e-91) {
		tmp = t_1;
	} else if ((x * y) <= 1.2e+49) {
		tmp = c + ((a * b) * -0.25);
	} else if ((x * y) <= 2.15e+102) {
		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 = c + (t * (z * 0.0625d0))
    if ((x * y) <= (-2.2d+168)) then
        tmp = x * y
    else if ((x * y) <= 7.6d-91) then
        tmp = t_1
    else if ((x * y) <= 1.2d+49) then
        tmp = c + ((a * b) * (-0.25d0))
    else if ((x * y) <= 2.15d+102) 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 = c + (t * (z * 0.0625));
	double tmp;
	if ((x * y) <= -2.2e+168) {
		tmp = x * y;
	} else if ((x * y) <= 7.6e-91) {
		tmp = t_1;
	} else if ((x * y) <= 1.2e+49) {
		tmp = c + ((a * b) * -0.25);
	} else if ((x * y) <= 2.15e+102) {
		tmp = t_1;
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (t * (z * 0.0625))
	tmp = 0
	if (x * y) <= -2.2e+168:
		tmp = x * y
	elif (x * y) <= 7.6e-91:
		tmp = t_1
	elif (x * y) <= 1.2e+49:
		tmp = c + ((a * b) * -0.25)
	elif (x * y) <= 2.15e+102:
		tmp = t_1
	else:
		tmp = c + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(t * Float64(z * 0.0625)))
	tmp = 0.0
	if (Float64(x * y) <= -2.2e+168)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 7.6e-91)
		tmp = t_1;
	elseif (Float64(x * y) <= 1.2e+49)
		tmp = Float64(c + Float64(Float64(a * b) * -0.25));
	elseif (Float64(x * y) <= 2.15e+102)
		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 = c + (t * (z * 0.0625));
	tmp = 0.0;
	if ((x * y) <= -2.2e+168)
		tmp = x * y;
	elseif ((x * y) <= 7.6e-91)
		tmp = t_1;
	elseif ((x * y) <= 1.2e+49)
		tmp = c + ((a * b) * -0.25);
	elseif ((x * y) <= 2.15e+102)
		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[(c + N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.2e+168], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 7.6e-91], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1.2e+49], N[(c + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2.15e+102], t$95$1, N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.2000000000000002e168

   1. Initial program 89.6%

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

      1. associate-+l- 89.6%

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

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-/l* 89.6%

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

      5. associate-/l* 89.6%

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

   3. Simplified 89.6%

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

      1. fma-udef 89.6%

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

      2. div-inv 89.6%

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

      3. clear-num 89.6%

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

      4. div-inv 89.6%

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

      5. metadata-eval 89.6%

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

   5. Applied egg-rr 89.6%

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

   6. Taylor expanded in x around inf 77.6%

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

   if -2.2000000000000002e168 < (*.f64 x y) < 7.59999999999999957e-91 or 1.2e49 < (*.f64 x y) < 2.15e102

   1. Initial program 98.0%

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

   2. Taylor expanded in z around inf 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r* 71.8%

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

   4. Simplified 71.8%

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

   if 7.59999999999999957e-91 < (*.f64 x y) < 1.2e49

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 70.1%

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

      1. *-commutative 70.1%

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

   4. Simplified 70.1%

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

   if 2.15e102 < (*.f64 x y)

   1. Initial program 90.9%

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

   2. Taylor expanded in x around inf 74.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+168}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.6 \cdot 10^{-91}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;x \cdot y \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 8: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;x \cdot y \leq -1.15 \cdot 10^{+164}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 0.004:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;c\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))))
   (if (<= (* x y) -1.15e+164)
     (* x y)
     (if (<= (* x y) 0.004)
       t_1
       (if (<= (* x y) 7.5e+25) c (if (<= (* x y) 1.2e+102) 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 * (z * t);
	double tmp;
	if ((x * y) <= -1.15e+164) {
		tmp = x * y;
	} else if ((x * y) <= 0.004) {
		tmp = t_1;
	} else if ((x * y) <= 7.5e+25) {
		tmp = c;
	} else if ((x * y) <= 1.2e+102) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0625d0 * (z * t)
    if ((x * y) <= (-1.15d+164)) then
        tmp = x * y
    else if ((x * y) <= 0.004d0) then
        tmp = t_1
    else if ((x * y) <= 7.5d+25) then
        tmp = c
    else if ((x * y) <= 1.2d+102) then
        tmp = t_1
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (z * t);
	double tmp;
	if ((x * y) <= -1.15e+164) {
		tmp = x * y;
	} else if ((x * y) <= 0.004) {
		tmp = t_1;
	} else if ((x * y) <= 7.5e+25) {
		tmp = c;
	} else if ((x * y) <= 1.2e+102) {
		tmp = t_1;
	} 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 (x * y) <= -1.15e+164:
		tmp = x * y
	elif (x * y) <= 0.004:
		tmp = t_1
	elif (x * y) <= 7.5e+25:
		tmp = c
	elif (x * y) <= 1.2e+102:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (Float64(x * y) <= -1.15e+164)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 0.004)
		tmp = t_1;
	elseif (Float64(x * y) <= 7.5e+25)
		tmp = c;
	elseif (Float64(x * y) <= 1.2e+102)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 0.0625 * (z * t);
	tmp = 0.0;
	if ((x * y) <= -1.15e+164)
		tmp = x * y;
	elseif ((x * y) <= 0.004)
		tmp = t_1;
	elseif ((x * y) <= 7.5e+25)
		tmp = c;
	elseif ((x * y) <= 1.2e+102)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1.15e+164], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.004], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 7.5e+25], c, If[LessEqual[N[(x * y), $MachinePrecision], 1.2e+102], t$95$1, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.15e164 or 1.19999999999999997e102 < (*.f64 x y)

   1. Initial program 90.4%

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

      1. associate-+l- 90.4%

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

      2. fma-def 93.1%

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

      3. *-commutative 93.1%

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

      4. associate-/l* 93.1%

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

      5. associate-/l* 93.1%

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

   3. Simplified 93.1%

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

      1. fma-udef 90.4%

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

      2. div-inv 90.4%

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

      3. clear-num 90.4%

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

      4. div-inv 90.4%

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

      5. metadata-eval 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in x around inf 72.7%

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

   if -1.15e164 < (*.f64 x y) < 0.0040000000000000001 or 7.49999999999999993e25 < (*.f64 x y) < 1.19999999999999997e102

   1. Initial program 98.2%

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

      1. associate-+l- 98.2%

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

      2. fma-def 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-/l* 98.1%

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

      5. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

      1. fma-udef 98.1%

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

      2. div-inv 98.1%

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

      3. clear-num 98.2%

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

      4. div-inv 98.2%

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

      5. metadata-eval 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in t around inf 43.6%

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

   if 0.0040000000000000001 < (*.f64 x y) < 7.49999999999999993e25

   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 c around inf 57.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.15 \cdot 10^{+164}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 0.004:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;x \cdot y \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;c\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{+102}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq 1.1 \cdot 10^{+79}\right):\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + t_1\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))))
   (if (or (<= (* x y) -4e+97) (not (<= (* x y) 1.1e+79)))
     (+ c (+ (* x y) t_1))
     (- (+ 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 = 0.0625 * (z * t);
	double tmp;
	if (((x * y) <= -4e+97) || !((x * y) <= 1.1e+79)) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (c + t_1) - ((a * b) * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.0625d0 * (z * t)
    if (((x * y) <= (-4d+97)) .or. (.not. ((x * y) <= 1.1d+79))) then
        tmp = c + ((x * y) + t_1)
    else
        tmp = (c + t_1) - ((a * b) * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (z * t);
	double tmp;
	if (((x * y) <= -4e+97) || !((x * y) <= 1.1e+79)) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (c + t_1) - ((a * b) * 0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 0.0625 * (z * t)
	tmp = 0
	if ((x * y) <= -4e+97) or not ((x * y) <= 1.1e+79):
		tmp = c + ((x * y) + t_1)
	else:
		tmp = (c + t_1) - ((a * b) * 0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if ((Float64(x * y) <= -4e+97) || !(Float64(x * y) <= 1.1e+79))
		tmp = Float64(c + Float64(Float64(x * y) + t_1));
	else
		tmp = Float64(Float64(c + t_1) - Float64(Float64(a * b) * 0.25));
	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 (((x * y) <= -4e+97) || ~(((x * y) <= 1.1e+79)))
		tmp = c + ((x * y) + t_1);
	else
		tmp = (c + t_1) - ((a * b) * 0.25);
	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[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+97], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1.1e+79]], $MachinePrecision]], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(c + t$95$1), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.0000000000000003e97 or 1.0999999999999999e79 < (*.f64 x y)

   1. Initial program 90.8%

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

   2. Taylor expanded in a around 0 83.5%

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

   if -4.0000000000000003e97 < (*.f64 x y) < 1.0999999999999999e79

   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 0 94.1%

      \[\leadsto \color{blue}{\left(c + 0.0625 \cdot \left(t \cdot z\right)\right) - 0.25 \cdot \left(a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq 1.1 \cdot 10^{+79}\right):\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + 0.0625 \cdot \left(z \cdot t\right)\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 10: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{+164}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* a b) -8.5e+133)
   (- (* x y) (* (* a b) 0.25))
   (if (<= (* a b) 4e+164)
     (+ c (+ (* x y) (* 0.0625 (* z t))))
     (+ c (* (* a b) -0.25)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a * b) <= -8.5e+133) {
		tmp = (x * y) - ((a * b) * 0.25);
	} else if ((a * b) <= 4e+164) {
		tmp = c + ((x * y) + (0.0625 * (z * t)));
	} else {
		tmp = c + ((a * b) * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((a * b) <= (-8.5d+133)) then
        tmp = (x * y) - ((a * b) * 0.25d0)
    else if ((a * b) <= 4d+164) then
        tmp = c + ((x * y) + (0.0625d0 * (z * t)))
    else
        tmp = c + ((a * b) * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a * b) <= -8.5e+133) {
		tmp = (x * y) - ((a * b) * 0.25);
	} else if ((a * b) <= 4e+164) {
		tmp = c + ((x * y) + (0.0625 * (z * t)));
	} else {
		tmp = c + ((a * b) * -0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (a * b) <= -8.5e+133:
		tmp = (x * y) - ((a * b) * 0.25)
	elif (a * b) <= 4e+164:
		tmp = c + ((x * y) + (0.0625 * (z * t)))
	else:
		tmp = c + ((a * b) * -0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(a * b) <= -8.5e+133)
		tmp = Float64(Float64(x * y) - Float64(Float64(a * b) * 0.25));
	elseif (Float64(a * b) <= 4e+164)
		tmp = Float64(c + Float64(Float64(x * y) + Float64(0.0625 * Float64(z * t))));
	else
		tmp = Float64(c + Float64(Float64(a * b) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((a * b) <= -8.5e+133)
		tmp = (x * y) - ((a * b) * 0.25);
	elseif ((a * b) <= 4e+164)
		tmp = c + ((x * y) + (0.0625 * (z * t)));
	else
		tmp = c + ((a * b) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(a * b), $MachinePrecision], -8.5e+133], N[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4e+164], N[(c + N[(N[(x * y), $MachinePrecision] + N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -8.50000000000000044e133

   1. Initial program 85.7%

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

   2. Taylor expanded in z around 0 83.2%

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

   3. Taylor expanded in c around 0 77.5%

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

   if -8.50000000000000044e133 < (*.f64 a b) < 4e164

   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 90.9%

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

   if 4e164 < (*.f64 a b)

   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 a around inf 85.5%

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

      1. *-commutative 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{+164}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \end{array} \]

Alternative 11: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+133}:\\ \;\;\;\;x \cdot y - t_1\\ \mathbf{elif}\;a \cdot b \leq 1.3 \cdot 10^{+179}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)) (t_2 (* 0.0625 (* z t))))
   (if (<= (* a b) -4e+133)
     (- (* x y) t_1)
     (if (<= (* a b) 1.3e+179) (+ c (+ (* x y) t_2)) (- t_2 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 t_2 = 0.0625 * (z * t);
	double tmp;
	if ((a * b) <= -4e+133) {
		tmp = (x * y) - t_1;
	} else if ((a * b) <= 1.3e+179) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = t_2 - 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) :: t_2
    real(8) :: tmp
    t_1 = (a * b) * 0.25d0
    t_2 = 0.0625d0 * (z * t)
    if ((a * b) <= (-4d+133)) then
        tmp = (x * y) - t_1
    else if ((a * b) <= 1.3d+179) then
        tmp = c + ((x * y) + t_2)
    else
        tmp = t_2 - 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 t_2 = 0.0625 * (z * t);
	double tmp;
	if ((a * b) <= -4e+133) {
		tmp = (x * y) - t_1;
	} else if ((a * b) <= 1.3e+179) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	t_2 = 0.0625 * (z * t)
	tmp = 0
	if (a * b) <= -4e+133:
		tmp = (x * y) - t_1
	elif (a * b) <= 1.3e+179:
		tmp = c + ((x * y) + t_2)
	else:
		tmp = t_2 - t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) * 0.25)
	t_2 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -4e+133)
		tmp = Float64(Float64(x * y) - t_1);
	elseif (Float64(a * b) <= 1.3e+179)
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	else
		tmp = Float64(t_2 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * 0.25;
	t_2 = 0.0625 * (z * t);
	tmp = 0.0;
	if ((a * b) <= -4e+133)
		tmp = (x * y) - t_1;
	elseif ((a * b) <= 1.3e+179)
		tmp = c + ((x * y) + t_2);
	else
		tmp = t_2 - 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]}, Block[{t$95$2 = N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -4e+133], N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.3e+179], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.0000000000000001e133

   1. Initial program 85.7%

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

   2. Taylor expanded in z around 0 83.2%

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

   3. Taylor expanded in c around 0 77.5%

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

   if -4.0000000000000001e133 < (*.f64 a b) < 1.3000000000000001e179

   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 91.0%

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

   if 1.3000000000000001e179 < (*.f64 a b)

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0 92.3%

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

   3. Taylor expanded in c around 0 92.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+133}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 1.3 \cdot 10^{+179}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 12: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;a \cdot b \leq -1.14 \cdot 10^{+133}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{+167}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)) (t_2 (* 0.0625 (* z t))))
   (if (<= (* a b) -1.14e+133)
     (- (+ c (* x y)) t_1)
     (if (<= (* a b) 1.8e+167) (+ c (+ (* x y) t_2)) (- t_2 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 t_2 = 0.0625 * (z * t);
	double tmp;
	if ((a * b) <= -1.14e+133) {
		tmp = (c + (x * y)) - t_1;
	} else if ((a * b) <= 1.8e+167) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = t_2 - 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) :: t_2
    real(8) :: tmp
    t_1 = (a * b) * 0.25d0
    t_2 = 0.0625d0 * (z * t)
    if ((a * b) <= (-1.14d+133)) then
        tmp = (c + (x * y)) - t_1
    else if ((a * b) <= 1.8d+167) then
        tmp = c + ((x * y) + t_2)
    else
        tmp = t_2 - 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 t_2 = 0.0625 * (z * t);
	double tmp;
	if ((a * b) <= -1.14e+133) {
		tmp = (c + (x * y)) - t_1;
	} else if ((a * b) <= 1.8e+167) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	t_2 = 0.0625 * (z * t)
	tmp = 0
	if (a * b) <= -1.14e+133:
		tmp = (c + (x * y)) - t_1
	elif (a * b) <= 1.8e+167:
		tmp = c + ((x * y) + t_2)
	else:
		tmp = t_2 - t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) * 0.25)
	t_2 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -1.14e+133)
		tmp = Float64(Float64(c + Float64(x * y)) - t_1);
	elseif (Float64(a * b) <= 1.8e+167)
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	else
		tmp = Float64(t_2 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * 0.25;
	t_2 = 0.0625 * (z * t);
	tmp = 0.0;
	if ((a * b) <= -1.14e+133)
		tmp = (c + (x * y)) - t_1;
	elseif ((a * b) <= 1.8e+167)
		tmp = c + ((x * y) + t_2);
	else
		tmp = t_2 - 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]}, Block[{t$95$2 = N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1.14e+133], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.8e+167], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.14e133

   1. Initial program 85.7%

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

   2. Taylor expanded in z around 0 83.2%

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

   if -1.14e133 < (*.f64 a b) < 1.80000000000000012e167

   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 91.0%

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

   if 1.80000000000000012e167 < (*.f64 a b)

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0 92.3%

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

   3. Taylor expanded in c around 0 92.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.14 \cdot 10^{+133}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{+167}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 13: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -6 \cdot 10^{+174}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -6e+174)
   (* x y)
   (if (<= (* x y) 6.2e+101) (+ c (* t (* z 0.0625))) (+ c (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -6e+174) {
		tmp = x * y;
	} else if ((x * y) <= 6.2e+101) {
		tmp = c + (t * (z * 0.0625));
	} 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 ((x * y) <= (-6d+174)) then
        tmp = x * y
    else if ((x * y) <= 6.2d+101) then
        tmp = c + (t * (z * 0.0625d0))
    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 ((x * y) <= -6e+174) {
		tmp = x * y;
	} else if ((x * y) <= 6.2e+101) {
		tmp = c + (t * (z * 0.0625));
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -6e+174:
		tmp = x * y
	elif (x * y) <= 6.2e+101:
		tmp = c + (t * (z * 0.0625))
	else:
		tmp = c + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -6e+174)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 6.2e+101)
		tmp = Float64(c + Float64(t * Float64(z * 0.0625)));
	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 ((x * y) <= -6e+174)
		tmp = x * y;
	elseif ((x * y) <= 6.2e+101)
		tmp = c + (t * (z * 0.0625));
	else
		tmp = c + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -6e+174], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6.2e+101], N[(c + N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -6e174

   1. Initial program 89.6%

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

      1. associate-+l- 89.6%

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

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-/l* 89.6%

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

      5. associate-/l* 89.6%

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

   3. Simplified 89.6%

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

      1. fma-udef 89.6%

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

      2. div-inv 89.6%

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

      3. clear-num 89.6%

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

      4. div-inv 89.6%

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

      5. metadata-eval 89.6%

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

   5. Applied egg-rr 89.6%

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

   6. Taylor expanded in x around inf 77.6%

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

   if -6e174 < (*.f64 x y) < 6.19999999999999998e101

   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 z around inf 69.2%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*r* 69.2%

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

   4. Simplified 69.2%

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

   if 6.19999999999999998e101 < (*.f64 x y)

   1. Initial program 90.9%

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

   2. Taylor expanded in x around inf 74.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -6 \cdot 10^{+174}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 14: 49.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + x \cdot y\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-106}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* x y))) (t_2 (* 0.0625 (* z t))))
   (if (<= z -3.4e+186)
     t_2
     (if (<= z -2.6e-50)
       t_1
       (if (<= z -3.8e-106) (* (* a b) -0.25) (if (<= z 2.2e-99) 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 + (x * y);
	double t_2 = 0.0625 * (z * t);
	double tmp;
	if (z <= -3.4e+186) {
		tmp = t_2;
	} else if (z <= -2.6e-50) {
		tmp = t_1;
	} else if (z <= -3.8e-106) {
		tmp = (a * b) * -0.25;
	} else if (z <= 2.2e-99) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = c + (x * y)
    t_2 = 0.0625d0 * (z * t)
    if (z <= (-3.4d+186)) then
        tmp = t_2
    else if (z <= (-2.6d-50)) then
        tmp = t_1
    else if (z <= (-3.8d-106)) then
        tmp = (a * b) * (-0.25d0)
    else if (z <= 2.2d-99) 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 + (x * y);
	double t_2 = 0.0625 * (z * t);
	double tmp;
	if (z <= -3.4e+186) {
		tmp = t_2;
	} else if (z <= -2.6e-50) {
		tmp = t_1;
	} else if (z <= -3.8e-106) {
		tmp = (a * b) * -0.25;
	} else if (z <= 2.2e-99) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (x * y)
	t_2 = 0.0625 * (z * t)
	tmp = 0
	if z <= -3.4e+186:
		tmp = t_2
	elif z <= -2.6e-50:
		tmp = t_1
	elif z <= -3.8e-106:
		tmp = (a * b) * -0.25
	elif z <= 2.2e-99:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(x * y))
	t_2 = Float64(0.0625 * Float64(z * t))
	tmp = 0.0
	if (z <= -3.4e+186)
		tmp = t_2;
	elseif (z <= -2.6e-50)
		tmp = t_1;
	elseif (z <= -3.8e-106)
		tmp = Float64(Float64(a * b) * -0.25);
	elseif (z <= 2.2e-99)
		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 + (x * y);
	t_2 = 0.0625 * (z * t);
	tmp = 0.0;
	if (z <= -3.4e+186)
		tmp = t_2;
	elseif (z <= -2.6e-50)
		tmp = t_1;
	elseif (z <= -3.8e-106)
		tmp = (a * b) * -0.25;
	elseif (z <= 2.2e-99)
		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[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+186], t$95$2, If[LessEqual[z, -2.6e-50], t$95$1, If[LessEqual[z, -3.8e-106], N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision], If[LessEqual[z, 2.2e-99], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.40000000000000005e186 or 2.20000000000000004e-99 < z

   1. Initial program 97.3%

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

      1. associate-+l- 97.3%

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

      2. fma-def 97.3%

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

      3. *-commutative 97.3%

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

      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* 97.1%

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

   3. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. div-inv 97.2%

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

      3. clear-num 97.3%

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

      4. div-inv 97.3%

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

      5. metadata-eval 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in t around inf 52.3%

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

   if -3.40000000000000005e186 < z < -2.6000000000000001e-50 or -3.7999999999999999e-106 < z < 2.20000000000000004e-99

   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 inf 58.6%

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

   if -2.6000000000000001e-50 < z < -3.7999999999999999e-106

   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* 99.8%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. div-inv 99.7%

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

      3. clear-num 99.9%

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

      4. div-inv 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around inf 41.7%

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

      1. *-commutative 41.7%

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

   8. Simplified 41.7%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+186}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-106}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]

Alternative 15: 41.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5.4 \cdot 10^{+101}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2.3e+82) (* x y) (if (<= (* x y) 5.4e+101) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -2.3e+82) {
		tmp = x * y;
	} else if ((x * y) <= 5.4e+101) {
		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 ((x * y) <= (-2.3d+82)) then
        tmp = x * y
    else if ((x * y) <= 5.4d+101) 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 ((x * y) <= -2.3e+82) {
		tmp = x * y;
	} else if ((x * y) <= 5.4e+101) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -2.3e+82:
		tmp = x * y
	elif (x * y) <= 5.4e+101:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2.3e+82)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 5.4e+101)
		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 ((x * y) <= -2.3e+82)
		tmp = x * y;
	elseif ((x * y) <= 5.4e+101)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.3e+82], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5.4e+101], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.29999999999999988e82 or 5.40000000000000012e101 < (*.f64 x y)

   1. Initial program 90.7%

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

      1. associate-+l- 90.7%

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

      2. fma-def 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-/l* 93.0%

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

      5. associate-/l* 93.0%

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

   3. Simplified 93.0%

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

      1. fma-udef 90.7%

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

      2. div-inv 90.6%

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

      3. clear-num 90.7%

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

      4. div-inv 90.7%

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

      5. metadata-eval 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in x around inf 64.4%

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

   if -2.29999999999999988e82 < (*.f64 x y) < 5.40000000000000012e101

   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 c around inf 31.1%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5.4 \cdot 10^{+101}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16: 22.2% 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 96.1%

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

2. Taylor expanded in c around inf 22.6%

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

3. Final simplification 22.6%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023279 
(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))