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

Percentage Accurate: 97.8% → 98.6%

Time: 11.5s

Alternatives: 13

Speedup: 0.5×

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

Specification

Math

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

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: 97.8% 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: 98.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) INFINITY)
   (+ c (- (+ (* x y) (* z (* t 0.0625))) (/ a (/ 4.0 b))))
   (+ 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) + ((z * t) / 16.0)) - ((a * b) / 4.0)) <= ((double) INFINITY)) {
		tmp = c + (((x * y) + (z * (t * 0.0625))) - (a / (4.0 / b)));
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) <= Double.POSITIVE_INFINITY) {
		tmp = c + (((x * y) + (z * (t * 0.0625))) - (a / (4.0 / b)));
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) <= math.inf:
		tmp = c + (((x * y) + (z * (t * 0.0625))) - (a / (4.0 / b)))
	else:
		tmp = c + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) <= Inf)
		tmp = Float64(c + Float64(Float64(Float64(x * y) + Float64(z * Float64(t * 0.0625))) - Float64(a / Float64(4.0 / b))));
	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) + ((z * t) / 16.0)) - ((a * b) / 4.0)) <= Inf)
		tmp = c + (((x * y) + (z * (t * 0.0625))) - (a / (4.0 / b)));
	else
		tmp = c + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

      4. associate-/l* 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. div-inv 99.6%

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

      5. associate-*l* 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   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 x around inf 63.7%

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

4. Final simplification 98.8%

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

Alternative 2: 98.9% 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 97.3%

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

   4. associate-*l/ 97.6%

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

   5. fma-neg 97.6%

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

   6. sub-neg 97.6%

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

   7. distribute-neg-in 97.6%

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

   8. remove-double-neg 97.6%

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

   9. associate-/l* 97.6%

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

   10. distribute-frac-neg 97.6%

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

   11. associate-/r/ 97.6%

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

   12. fma-def 97.6%

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

   13. neg-mul-1 97.6%

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

   14. *-commutative 97.6%

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

   15. associate-/l* 97.6%

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

   16. metadata-eval 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

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) (+ c (* 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 + (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 = c + (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 = c + (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 + 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 = c + (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[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   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 x around inf 63.7%

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

4. Final simplification 98.5%

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

Alternative 4: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + b \cdot \left(a \cdot -0.25\right)\\ t_2 := c + x \cdot y\\ \mathbf{if}\;x \cdot y \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;x \cdot y \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* b (* a -0.25)))) (t_2 (+ c (* x y))))
   (if (<= (* x y) -1.5e+79)
     t_2
     (if (<= (* x y) 4.4e-103)
       t_1
       (if (<= (* x y) 3.8e-39)
         (* t (* z 0.0625))
         (if (<= (* x y) 1.5e+134) 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 + (b * (a * -0.25));
	double t_2 = c + (x * y);
	double tmp;
	if ((x * y) <= -1.5e+79) {
		tmp = t_2;
	} else if ((x * y) <= 4.4e-103) {
		tmp = t_1;
	} else if ((x * y) <= 3.8e-39) {
		tmp = t * (z * 0.0625);
	} else if ((x * y) <= 1.5e+134) {
		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 + (b * (a * (-0.25d0)))
    t_2 = c + (x * y)
    if ((x * y) <= (-1.5d+79)) then
        tmp = t_2
    else if ((x * y) <= 4.4d-103) then
        tmp = t_1
    else if ((x * y) <= 3.8d-39) then
        tmp = t * (z * 0.0625d0)
    else if ((x * y) <= 1.5d+134) 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 + (b * (a * -0.25));
	double t_2 = c + (x * y);
	double tmp;
	if ((x * y) <= -1.5e+79) {
		tmp = t_2;
	} else if ((x * y) <= 4.4e-103) {
		tmp = t_1;
	} else if ((x * y) <= 3.8e-39) {
		tmp = t * (z * 0.0625);
	} else if ((x * y) <= 1.5e+134) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = c + (b * (a * -0.25))
	t_2 = c + (x * y)
	tmp = 0
	if (x * y) <= -1.5e+79:
		tmp = t_2
	elif (x * y) <= 4.4e-103:
		tmp = t_1
	elif (x * y) <= 3.8e-39:
		tmp = t * (z * 0.0625)
	elif (x * y) <= 1.5e+134:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(b * Float64(a * -0.25)))
	t_2 = Float64(c + Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -1.5e+79)
		tmp = t_2;
	elseif (Float64(x * y) <= 4.4e-103)
		tmp = t_1;
	elseif (Float64(x * y) <= 3.8e-39)
		tmp = Float64(t * Float64(z * 0.0625));
	elseif (Float64(x * y) <= 1.5e+134)
		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 + (b * (a * -0.25));
	t_2 = c + (x * y);
	tmp = 0.0;
	if ((x * y) <= -1.5e+79)
		tmp = t_2;
	elseif ((x * y) <= 4.4e-103)
		tmp = t_1;
	elseif ((x * y) <= 3.8e-39)
		tmp = t * (z * 0.0625);
	elseif ((x * y) <= 1.5e+134)
		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[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1.5e+79], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 4.4e-103], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 3.8e-39], N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.5e+134], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.49999999999999987e79 or 1.49999999999999998e134 < (*.f64 x y)

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 77.2%

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

   if -1.49999999999999987e79 < (*.f64 x y) < 4.3999999999999999e-103 or 3.8000000000000002e-39 < (*.f64 x y) < 1.49999999999999998e134

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

      3. *-commutative 62.6%

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

   4. Simplified 62.6%

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

   if 4.3999999999999999e-103 < (*.f64 x y) < 3.8000000000000002e-39

   1. Initial program 91.7%

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

   2. Taylor expanded in x around 0 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. metadata-eval 91.7%

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

      3. +-commutative 91.7%

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

      4. associate-*r* 91.7%

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

      5. *-commutative 91.7%

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

      6. *-commutative 91.7%

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

      7. *-commutative 91.7%

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

      8. *-commutative 91.7%

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

      9. associate-*r* 91.7%

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

      10. fma-def 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-*l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 76.3%

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

   7. Simplified 76.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;x \cdot y \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 5: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + t \cdot \left(z \cdot 0.0625\right)\\ t_2 := c + x \cdot y\\ \mathbf{if}\;x \cdot y \leq -5.6 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 5.8 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;x \cdot y \leq 1.5 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* t (* z 0.0625)))) (t_2 (+ c (* x y))))
   (if (<= (* x y) -5.6e+78)
     t_2
     (if (<= (* x y) 5.8e-192)
       t_1
       (if (<= (* x y) 5.5e-105)
         (+ c (* b (* a -0.25)))
         (if (<= (* x y) 1.5e+45) 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 + (x * y);
	double tmp;
	if ((x * y) <= -5.6e+78) {
		tmp = t_2;
	} else if ((x * y) <= 5.8e-192) {
		tmp = t_1;
	} else if ((x * y) <= 5.5e-105) {
		tmp = c + (b * (a * -0.25));
	} else if ((x * y) <= 1.5e+45) {
		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 = c + (x * y)
    if ((x * y) <= (-5.6d+78)) then
        tmp = t_2
    else if ((x * y) <= 5.8d-192) then
        tmp = t_1
    else if ((x * y) <= 5.5d-105) then
        tmp = c + (b * (a * (-0.25d0)))
    else if ((x * y) <= 1.5d+45) 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 + (x * y);
	double tmp;
	if ((x * y) <= -5.6e+78) {
		tmp = t_2;
	} else if ((x * y) <= 5.8e-192) {
		tmp = t_1;
	} else if ((x * y) <= 5.5e-105) {
		tmp = c + (b * (a * -0.25));
	} else if ((x * y) <= 1.5e+45) {
		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 + (x * y)
	tmp = 0
	if (x * y) <= -5.6e+78:
		tmp = t_2
	elif (x * y) <= 5.8e-192:
		tmp = t_1
	elif (x * y) <= 5.5e-105:
		tmp = c + (b * (a * -0.25))
	elif (x * y) <= 1.5e+45:
		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(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5.6e+78)
		tmp = t_2;
	elseif (Float64(x * y) <= 5.8e-192)
		tmp = t_1;
	elseif (Float64(x * y) <= 5.5e-105)
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	elseif (Float64(x * y) <= 1.5e+45)
		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 + (x * y);
	tmp = 0.0;
	if ((x * y) <= -5.6e+78)
		tmp = t_2;
	elseif ((x * y) <= 5.8e-192)
		tmp = t_1;
	elseif ((x * y) <= 5.5e-105)
		tmp = c + (b * (a * -0.25));
	elseif ((x * y) <= 1.5e+45)
		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[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5.6e+78], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 5.8e-192], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5.5e-105], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.5e+45], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.6000000000000002e78 or 1.50000000000000005e45 < (*.f64 x y)

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf 70.2%

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

   if -5.6000000000000002e78 < (*.f64 x y) < 5.80000000000000033e-192 or 5.50000000000000029e-105 < (*.f64 x y) < 1.50000000000000005e45

   1. Initial program 99.3%

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

   2. Taylor expanded in z around inf 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*r* 70.2%

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

      3. *-commutative 70.2%

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

   4. Simplified 70.2%

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

   if 5.80000000000000033e-192 < (*.f64 x y) < 5.50000000000000029e-105

   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 86.9%

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

      1. associate-*r* 86.9%

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

      2. *-commutative 86.9%

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

      3. *-commutative 86.9%

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

   4. Simplified 86.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.6 \cdot 10^{+78}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5.8 \cdot 10^{-192}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;x \cdot y \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;x \cdot y \leq 1.5 \cdot 10^{+45}:\\ \;\;\;\;c + t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 6: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := \left(z \cdot t\right) \cdot 0.0625\\ t_3 := c + \left(x \cdot y - t_1\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-38}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+196}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25))
        (t_2 (* (* z t) 0.0625))
        (t_3 (+ c (- (* x y) t_1))))
   (if (<= (* a b) -2e+116)
     t_3
     (if (<= (* a b) 5e-38)
       (+ c (+ (* x y) t_2))
       (if (<= (* a b) 5e+196) t_3 (- 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 = (z * t) * 0.0625;
	double t_3 = c + ((x * y) - t_1);
	double tmp;
	if ((a * b) <= -2e+116) {
		tmp = t_3;
	} else if ((a * b) <= 5e-38) {
		tmp = c + ((x * y) + t_2);
	} else if ((a * b) <= 5e+196) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = (a * b) * 0.25d0
    t_2 = (z * t) * 0.0625d0
    t_3 = c + ((x * y) - t_1)
    if ((a * b) <= (-2d+116)) then
        tmp = t_3
    else if ((a * b) <= 5d-38) then
        tmp = c + ((x * y) + t_2)
    else if ((a * b) <= 5d+196) then
        tmp = t_3
    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 = (z * t) * 0.0625;
	double t_3 = c + ((x * y) - t_1);
	double tmp;
	if ((a * b) <= -2e+116) {
		tmp = t_3;
	} else if ((a * b) <= 5e-38) {
		tmp = c + ((x * y) + t_2);
	} else if ((a * b) <= 5e+196) {
		tmp = t_3;
	} 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 = (z * t) * 0.0625
	t_3 = c + ((x * y) - t_1)
	tmp = 0
	if (a * b) <= -2e+116:
		tmp = t_3
	elif (a * b) <= 5e-38:
		tmp = c + ((x * y) + t_2)
	elif (a * b) <= 5e+196:
		tmp = t_3
	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(Float64(z * t) * 0.0625)
	t_3 = Float64(c + Float64(Float64(x * y) - t_1))
	tmp = 0.0
	if (Float64(a * b) <= -2e+116)
		tmp = t_3;
	elseif (Float64(a * b) <= 5e-38)
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	elseif (Float64(a * b) <= 5e+196)
		tmp = t_3;
	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 = (z * t) * 0.0625;
	t_3 = c + ((x * y) - t_1);
	tmp = 0.0;
	if ((a * b) <= -2e+116)
		tmp = t_3;
	elseif ((a * b) <= 5e-38)
		tmp = c + ((x * y) + t_2);
	elseif ((a * b) <= 5e+196)
		tmp = t_3;
	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[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(c + N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+116], t$95$3, If[LessEqual[N[(a * b), $MachinePrecision], 5e-38], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+196], t$95$3, N[(t$95$2 - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 5.00000000000000033e-38 < (*.f64 a b) < 4.9999999999999998e196

   1. Initial program 93.6%

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

   2. Taylor expanded in z around 0 82.2%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 5.00000000000000033e-38

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 96.2%

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

   if 4.9999999999999998e196 < (*.f64 a b)

   1. Initial program 88.0%

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

      1. +-commutative 88.0%

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

      2. +-commutative 88.0%

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

      3. fma-def 92.0%

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

      4. associate-/l* 91.9%

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

      5. associate-/l* 91.7%

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

   3. Simplified 91.7%

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

      1. fma-udef 87.7%

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

      2. associate-/l* 87.8%

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

      3. +-commutative 87.8%

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

      4. div-inv 87.8%

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

      5. associate-*l* 87.8%

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

      6. metadata-eval 87.8%

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

   5. Applied egg-rr 87.8%

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

   6. Taylor expanded in x around 0 96.0%

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

   7. Taylor expanded in c around 0 96.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116}:\\ \;\;\;\;c + \left(x \cdot y - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-38}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+196}:\\ \;\;\;\;c + \left(x \cdot y - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625 - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 7: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c + x \cdot y\\ t_2 := \left(a \cdot b\right) \cdot -0.25\\ \mathbf{if}\;a \cdot b \leq -1.35 \cdot 10^{+115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -2.45 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -1.7 \cdot 10^{-199}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* x y))) (t_2 (* (* a b) -0.25)))
   (if (<= (* a b) -1.35e+115)
     t_2
     (if (<= (* a b) -2.45e-113)
       t_1
       (if (<= (* a b) -1.7e-199)
         (* t (* z 0.0625))
         (if (<= (* a b) 6.5e+76) 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 = (a * b) * -0.25;
	double tmp;
	if ((a * b) <= -1.35e+115) {
		tmp = t_2;
	} else if ((a * b) <= -2.45e-113) {
		tmp = t_1;
	} else if ((a * b) <= -1.7e-199) {
		tmp = t * (z * 0.0625);
	} else if ((a * b) <= 6.5e+76) {
		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 = (a * b) * (-0.25d0)
    if ((a * b) <= (-1.35d+115)) then
        tmp = t_2
    else if ((a * b) <= (-2.45d-113)) then
        tmp = t_1
    else if ((a * b) <= (-1.7d-199)) then
        tmp = t * (z * 0.0625d0)
    else if ((a * b) <= 6.5d+76) 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 = (a * b) * -0.25;
	double tmp;
	if ((a * b) <= -1.35e+115) {
		tmp = t_2;
	} else if ((a * b) <= -2.45e-113) {
		tmp = t_1;
	} else if ((a * b) <= -1.7e-199) {
		tmp = t * (z * 0.0625);
	} else if ((a * b) <= 6.5e+76) {
		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 = (a * b) * -0.25
	tmp = 0
	if (a * b) <= -1.35e+115:
		tmp = t_2
	elif (a * b) <= -2.45e-113:
		tmp = t_1
	elif (a * b) <= -1.7e-199:
		tmp = t * (z * 0.0625)
	elif (a * b) <= 6.5e+76:
		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(Float64(a * b) * -0.25)
	tmp = 0.0
	if (Float64(a * b) <= -1.35e+115)
		tmp = t_2;
	elseif (Float64(a * b) <= -2.45e-113)
		tmp = t_1;
	elseif (Float64(a * b) <= -1.7e-199)
		tmp = Float64(t * Float64(z * 0.0625));
	elseif (Float64(a * b) <= 6.5e+76)
		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 = (a * b) * -0.25;
	tmp = 0.0;
	if ((a * b) <= -1.35e+115)
		tmp = t_2;
	elseif ((a * b) <= -2.45e-113)
		tmp = t_1;
	elseif ((a * b) <= -1.7e-199)
		tmp = t * (z * 0.0625);
	elseif ((a * b) <= 6.5e+76)
		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[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1.35e+115], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], -2.45e-113], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], -1.7e-199], N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 6.5e+76], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.35000000000000002e115 or 6.5000000000000005e76 < (*.f64 a b)

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0 86.4%

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

      1. cancel-sign-sub-inv 86.4%

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

      2. metadata-eval 86.4%

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

      3. +-commutative 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

      7. *-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-*r* 86.4%

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

      10. fma-def 87.6%

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

      11. *-commutative 87.6%

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

      12. associate-*l* 87.6%

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

   4. Simplified 87.6%

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

   5. Taylor expanded in b around inf 64.9%

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

   if -1.35000000000000002e115 < (*.f64 a b) < -2.4500000000000001e-113 or -1.70000000000000003e-199 < (*.f64 a b) < 6.5000000000000005e76

   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 64.4%

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

   if -2.4500000000000001e-113 < (*.f64 a b) < -1.70000000000000003e-199

   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 x around 0 87.0%

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

      1. cancel-sign-sub-inv 87.0%

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

      2. metadata-eval 87.0%

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

      3. +-commutative 87.0%

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

      4. associate-*r* 87.0%

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

      5. *-commutative 87.0%

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

      6. *-commutative 87.0%

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

      7. *-commutative 87.0%

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

      8. *-commutative 87.0%

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

      9. associate-*r* 87.0%

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

      10. fma-def 87.0%

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

      11. *-commutative 87.0%

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

      12. associate-*l* 87.0%

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

   4. Simplified 87.0%

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

   5. Taylor expanded in t around inf 73.7%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.35 \cdot 10^{+115}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;a \cdot b \leq -2.45 \cdot 10^{-113}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -1.7 \cdot 10^{-199}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \end{array} \]

Alternative 8: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+56} \lor \neg \left(a \cdot b \leq 10^{+38}\right):\\ \;\;\;\;c + \left(t_1 - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)))
   (if (or (<= (* a b) -4e+56) (not (<= (* a b) 1e+38)))
     (+ c (- t_1 (* (* a b) 0.25)))
     (+ 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 = (z * t) * 0.0625;
	double tmp;
	if (((a * b) <= -4e+56) || !((a * b) <= 1e+38)) {
		tmp = c + (t_1 - ((a * b) * 0.25));
	} 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 = (z * t) * 0.0625d0
    if (((a * b) <= (-4d+56)) .or. (.not. ((a * b) <= 1d+38))) then
        tmp = c + (t_1 - ((a * b) * 0.25d0))
    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 = (z * t) * 0.0625;
	double tmp;
	if (((a * b) <= -4e+56) || !((a * b) <= 1e+38)) {
		tmp = c + (t_1 - ((a * b) * 0.25));
	} else {
		tmp = c + ((x * y) + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (z * t) * 0.0625
	tmp = 0
	if ((a * b) <= -4e+56) or not ((a * b) <= 1e+38):
		tmp = c + (t_1 - ((a * b) * 0.25))
	else:
		tmp = c + ((x * y) + t_1)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	tmp = 0.0
	if ((Float64(a * b) <= -4e+56) || !(Float64(a * b) <= 1e+38))
		tmp = Float64(c + Float64(t_1 - Float64(Float64(a * b) * 0.25)));
	else
		tmp = Float64(c + Float64(Float64(x * y) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) * 0.0625;
	tmp = 0.0;
	if (((a * b) <= -4e+56) || ~(((a * b) <= 1e+38)))
		tmp = c + (t_1 - ((a * b) * 0.25));
	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[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[Or[LessEqual[N[(a * b), $MachinePrecision], -4e+56], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+38]], $MachinePrecision]], N[(c + N[(t$95$1 - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.00000000000000037e56 or 9.99999999999999977e37 < (*.f64 a b)

   1. Initial program 93.1%

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

   2. Taylor expanded in x around 0 85.2%

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

   if -4.00000000000000037e56 < (*.f64 a b) < 9.99999999999999977e37

   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 a around 0 96.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+56} \lor \neg \left(a \cdot b \leq 10^{+38}\right):\\ \;\;\;\;c + \left(\left(z \cdot t\right) \cdot 0.0625 - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 9: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116} \lor \neg \left(a \cdot b \leq 10^{+77}\right):\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -2e+116) (not (<= (* a b) 1e+77)))
   (+ c (* b (* a -0.25)))
   (+ c (+ (* x y) (* (* z t) 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -2e+116) || !((a * b) <= 1e+77)) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = c + ((x * y) + ((z * t) * 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 (((a * b) <= (-2d+116)) .or. (.not. ((a * b) <= 1d+77))) then
        tmp = c + (b * (a * (-0.25d0)))
    else
        tmp = c + ((x * y) + ((z * t) * 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 (((a * b) <= -2e+116) || !((a * b) <= 1e+77)) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((a * b) <= -2e+116) or not ((a * b) <= 1e+77):
		tmp = c + (b * (a * -0.25))
	else:
		tmp = c + ((x * y) + ((z * t) * 0.0625))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+116) || !(Float64(a * b) <= 1e+77))
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	else
		tmp = Float64(c + Float64(Float64(x * y) + Float64(Float64(z * t) * 0.0625)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((a * b) <= -2e+116) || ~(((a * b) <= 1e+77)))
		tmp = c + (b * (a * -0.25));
	else
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+116], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+77]], $MachinePrecision]], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 9.99999999999999983e76 < (*.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 70.4%

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

      1. associate-*r* 70.4%

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

      2. *-commutative 70.4%

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

      3. *-commutative 70.4%

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

   4. Simplified 70.4%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 9.99999999999999983e76

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 93.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116} \lor \neg \left(a \cdot b \leq 10^{+77}\right):\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 10: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{-38}\right):\\ \;\;\;\;c + \left(x \cdot y - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -2e+116) (not (<= (* a b) 5e-38)))
   (+ c (- (* x y) (* (* a b) 0.25)))
   (+ c (+ (* x y) (* (* z t) 0.0625)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -2e+116) || !((a * b) <= 5e-38)) {
		tmp = c + ((x * y) - ((a * b) * 0.25));
	} else {
		tmp = c + ((x * y) + ((z * t) * 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 (((a * b) <= (-2d+116)) .or. (.not. ((a * b) <= 5d-38))) then
        tmp = c + ((x * y) - ((a * b) * 0.25d0))
    else
        tmp = c + ((x * y) + ((z * t) * 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 (((a * b) <= -2e+116) || !((a * b) <= 5e-38)) {
		tmp = c + ((x * y) - ((a * b) * 0.25));
	} else {
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((a * b) <= -2e+116) or not ((a * b) <= 5e-38):
		tmp = c + ((x * y) - ((a * b) * 0.25))
	else:
		tmp = c + ((x * y) + ((z * t) * 0.0625))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+116) || !(Float64(a * b) <= 5e-38))
		tmp = Float64(c + Float64(Float64(x * y) - Float64(Float64(a * b) * 0.25)));
	else
		tmp = Float64(c + Float64(Float64(x * y) + Float64(Float64(z * t) * 0.0625)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((a * b) <= -2e+116) || ~(((a * b) <= 5e-38)))
		tmp = c + ((x * y) - ((a * b) * 0.25));
	else
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+116], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e-38]], $MachinePrecision]], N[(c + N[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c + N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 5.00000000000000033e-38 < (*.f64 a b)

   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 z around 0 80.7%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 5.00000000000000033e-38

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 96.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{-38}\right):\\ \;\;\;\;c + \left(x \cdot y - \left(a \cdot b\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 11: 42.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot -0.25\\ \mathbf{if}\;a \cdot b \leq -2.55 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -8.4 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) -0.25)))
   (if (<= (* a b) -2.55e+92)
     t_1
     (if (<= (* a b) -8.4e-201)
       (* t (* z 0.0625))
       (if (<= (* a b) 2.3e-13) c 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 ((a * b) <= -2.55e+92) {
		tmp = t_1;
	} else if ((a * b) <= -8.4e-201) {
		tmp = t * (z * 0.0625);
	} else if ((a * b) <= 2.3e-13) {
		tmp = c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) * (-0.25d0)
    if ((a * b) <= (-2.55d+92)) then
        tmp = t_1
    else if ((a * b) <= (-8.4d-201)) then
        tmp = t * (z * 0.0625d0)
    else if ((a * b) <= 2.3d-13) then
        tmp = c
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) * -0.25;
	double tmp;
	if ((a * b) <= -2.55e+92) {
		tmp = t_1;
	} else if ((a * b) <= -8.4e-201) {
		tmp = t * (z * 0.0625);
	} else if ((a * b) <= 2.3e-13) {
		tmp = c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * -0.25
	tmp = 0
	if (a * b) <= -2.55e+92:
		tmp = t_1
	elif (a * b) <= -8.4e-201:
		tmp = t * (z * 0.0625)
	elif (a * b) <= 2.3e-13:
		tmp = c
	else:
		tmp = 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 (Float64(a * b) <= -2.55e+92)
		tmp = t_1;
	elseif (Float64(a * b) <= -8.4e-201)
		tmp = Float64(t * Float64(z * 0.0625));
	elseif (Float64(a * b) <= 2.3e-13)
		tmp = c;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * -0.25;
	tmp = 0.0;
	if ((a * b) <= -2.55e+92)
		tmp = t_1;
	elseif ((a * b) <= -8.4e-201)
		tmp = t * (z * 0.0625);
	elseif ((a * b) <= 2.3e-13)
		tmp = c;
	else
		tmp = 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[LessEqual[N[(a * b), $MachinePrecision], -2.55e+92], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], -8.4e-201], N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.3e-13], c, t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.5500000000000001e92 or 2.29999999999999979e-13 < (*.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 81.4%

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

      1. cancel-sign-sub-inv 81.4%

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

      2. metadata-eval 81.4%

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

      3. +-commutative 81.4%

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

      4. associate-*r* 81.4%

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

      5. *-commutative 81.4%

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

      6. *-commutative 81.4%

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

      7. *-commutative 81.4%

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

      8. *-commutative 81.4%

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

      9. associate-*r* 81.4%

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

      10. fma-def 82.4%

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

      11. *-commutative 82.4%

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

      12. associate-*l* 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in b around inf 56.5%

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

   if -2.5500000000000001e92 < (*.f64 a b) < -8.40000000000000049e-201

   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 x around 0 75.4%

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

      1. cancel-sign-sub-inv 75.4%

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

      2. metadata-eval 75.4%

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

      3. +-commutative 75.4%

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

      4. associate-*r* 75.4%

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

      5. *-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. *-commutative 75.4%

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

      8. *-commutative 75.4%

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

      9. associate-*r* 75.4%

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

      10. fma-def 75.4%

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

      11. *-commutative 75.4%

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

      12. associate-*l* 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in t around inf 51.7%

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

      1. associate-*r* 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-*r* 51.7%

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

   7. Simplified 51.7%

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

   if -8.40000000000000049e-201 < (*.f64 a b) < 2.29999999999999979e-13

   1. Initial program 99.1%

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

      1. associate-+l- 99.1%

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

      2. associate--l+ 99.1%

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

      3. fma-def 99.1%

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

      4. associate-*l/ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. associate-/l* 100.0%

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

      10. distribute-frac-neg 100.0%

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

      11. associate-/r/ 100.0%

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

      12. fma-def 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in c around inf 37.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.55 \cdot 10^{+92}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;a \cdot b \leq -8.4 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \end{array} \]

Alternative 12: 41.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.2 \cdot 10^{+36} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -3.2e+36) (not (<= (* a b) 2e-16))) (* (* a b) -0.25) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -3.2e+36) || !((a * b) <= 2e-16)) {
		tmp = (a * b) * -0.25;
	} else {
		tmp = c;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((a * b) <= (-3.2d+36)) .or. (.not. ((a * b) <= 2d-16))) then
        tmp = (a * b) * (-0.25d0)
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -3.2e+36) || !((a * b) <= 2e-16)) {
		tmp = (a * b) * -0.25;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((a * b) <= -3.2e+36) or not ((a * b) <= 2e-16):
		tmp = (a * b) * -0.25
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -3.2e+36) || !(Float64(a * b) <= 2e-16))
		tmp = Float64(Float64(a * b) * -0.25);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((a * b) <= -3.2e+36) || ~(((a * b) <= 2e-16)))
		tmp = (a * b) * -0.25;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -3.2e+36], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2e-16]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] * -0.25), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.1999999999999999e36 or 2e-16 < (*.f64 a b)

   1. Initial program 93.1%

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

   2. Taylor expanded in x around 0 81.7%

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

      1. cancel-sign-sub-inv 81.7%

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

      2. metadata-eval 81.7%

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

      3. +-commutative 81.7%

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

      4. associate-*r* 81.7%

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

      5. *-commutative 81.7%

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

      6. *-commutative 81.7%

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

      7. *-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. associate-*r* 81.7%

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

      10. fma-def 82.6%

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

      11. *-commutative 82.6%

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

      12. associate-*l* 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in b around inf 53.6%

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

   if -3.1999999999999999e36 < (*.f64 a b) < 2e-16

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.4%

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

      3. fma-def 99.4%

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

      4. associate-*l/ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. associate-/l* 100.0%

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

      10. distribute-frac-neg 100.0%

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

      11. associate-/r/ 100.0%

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

      12. fma-def 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in c around inf 31.9%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.2 \cdot 10^{+36} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 13: 21.7% 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. 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 97.3%

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

   4. associate-*l/ 97.6%

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

   5. fma-neg 97.6%

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

   6. sub-neg 97.6%

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

   7. distribute-neg-in 97.6%

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

   8. remove-double-neg 97.6%

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

   9. associate-/l* 97.6%

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

   10. distribute-frac-neg 97.6%

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

   11. associate-/r/ 97.6%

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

   12. fma-def 97.6%

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

   13. neg-mul-1 97.6%

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

   14. *-commutative 97.6%

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

   15. associate-/l* 97.6%

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

   16. metadata-eval 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in c around inf 22.7%

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

5. Final simplification 22.7%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023320 
(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))