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

Percentage Accurate: 97.7% → 98.1%

Time: 9.9s

Alternatives: 17

Speedup: 1.0×

• 32.1% 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: 97.7% 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: 98.1% accurate, 0.1× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (+ (fma x y (/ z (/ 16.0 t))) (- c (/ a (/ 4.0 b)))))

C

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

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(x * y + N[(z / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c - N[(a / N[(4.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. associate-+l- 98.0%

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

   2. sub-neg 98.0%

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

   3. neg-mul-1 98.0%

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

   4. metadata-eval 98.0%

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

   5. metadata-eval 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

   7. fma-def 99.2%

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

   8. associate-/l* 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-lft-identity 99.1%

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

   11. associate-/l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ t_2 := c + t_1\\ t_3 := x \cdot y + t_1\\ t_4 := x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-94}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-281}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625))
        (t_2 (+ c t_1))
        (t_3 (+ (* x y) t_1))
        (t_4 (- (* x y) (* (* a b) 0.25))))
   (if (<= (* a b) -5e+104)
     t_4
     (if (<= (* a b) -1e-94)
       (+ c (* x y))
       (if (<= (* a b) 2e-281)
         t_3
         (if (<= (* a b) 4e-116)
           t_2
           (if (<= (* a b) 5e-32) t_3 (if (<= (* a b) 1e+20) t_2 t_4))))))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) * 0.0625;
	double t_2 = c + t_1;
	double t_3 = (x * y) + t_1;
	double t_4 = (x * y) - ((a * b) * 0.25);
	double tmp;
	if ((a * b) <= -5e+104) {
		tmp = t_4;
	} else if ((a * b) <= -1e-94) {
		tmp = c + (x * y);
	} else if ((a * b) <= 2e-281) {
		tmp = t_3;
	} else if ((a * b) <= 4e-116) {
		tmp = t_2;
	} else if ((a * b) <= 5e-32) {
		tmp = t_3;
	} else if ((a * b) <= 1e+20) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (z * t) * 0.0625d0
    t_2 = c + t_1
    t_3 = (x * y) + t_1
    t_4 = (x * y) - ((a * b) * 0.25d0)
    if ((a * b) <= (-5d+104)) then
        tmp = t_4
    else if ((a * b) <= (-1d-94)) then
        tmp = c + (x * y)
    else if ((a * b) <= 2d-281) then
        tmp = t_3
    else if ((a * b) <= 4d-116) then
        tmp = t_2
    else if ((a * b) <= 5d-32) then
        tmp = t_3
    else if ((a * b) <= 1d+20) then
        tmp = t_2
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

assert a < b;
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 t_2 = c + t_1;
	double t_3 = (x * y) + t_1;
	double t_4 = (x * y) - ((a * b) * 0.25);
	double tmp;
	if ((a * b) <= -5e+104) {
		tmp = t_4;
	} else if ((a * b) <= -1e-94) {
		tmp = c + (x * y);
	} else if ((a * b) <= 2e-281) {
		tmp = t_3;
	} else if ((a * b) <= 4e-116) {
		tmp = t_2;
	} else if ((a * b) <= 5e-32) {
		tmp = t_3;
	} else if ((a * b) <= 1e+20) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (z * t) * 0.0625
	t_2 = c + t_1
	t_3 = (x * y) + t_1
	t_4 = (x * y) - ((a * b) * 0.25)
	tmp = 0
	if (a * b) <= -5e+104:
		tmp = t_4
	elif (a * b) <= -1e-94:
		tmp = c + (x * y)
	elif (a * b) <= 2e-281:
		tmp = t_3
	elif (a * b) <= 4e-116:
		tmp = t_2
	elif (a * b) <= 5e-32:
		tmp = t_3
	elif (a * b) <= 1e+20:
		tmp = t_2
	else:
		tmp = t_4
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	t_2 = Float64(c + t_1)
	t_3 = Float64(Float64(x * y) + t_1)
	t_4 = Float64(Float64(x * y) - Float64(Float64(a * b) * 0.25))
	tmp = 0.0
	if (Float64(a * b) <= -5e+104)
		tmp = t_4;
	elseif (Float64(a * b) <= -1e-94)
		tmp = Float64(c + Float64(x * y));
	elseif (Float64(a * b) <= 2e-281)
		tmp = t_3;
	elseif (Float64(a * b) <= 4e-116)
		tmp = t_2;
	elseif (Float64(a * b) <= 5e-32)
		tmp = t_3;
	elseif (Float64(a * b) <= 1e+20)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) * 0.0625;
	t_2 = c + t_1;
	t_3 = (x * y) + t_1;
	t_4 = (x * y) - ((a * b) * 0.25);
	tmp = 0.0;
	if ((a * b) <= -5e+104)
		tmp = t_4;
	elseif ((a * b) <= -1e-94)
		tmp = c + (x * y);
	elseif ((a * b) <= 2e-281)
		tmp = t_3;
	elseif ((a * b) <= 4e-116)
		tmp = t_2;
	elseif ((a * b) <= 5e-32)
		tmp = t_3;
	elseif ((a * b) <= 1e+20)
		tmp = t_2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(c + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+104], t$95$4, If[LessEqual[N[(a * b), $MachinePrecision], -1e-94], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e-281], t$95$3, If[LessEqual[N[(a * b), $MachinePrecision], 4e-116], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], 5e-32], t$95$3, If[LessEqual[N[(a * b), $MachinePrecision], 1e+20], t$95$2, t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -4.9999999999999997e104 or 1e20 < (*.f64 a b)

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0 86.6%

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

   3. Taylor expanded in c around 0 82.4%

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

   if -4.9999999999999997e104 < (*.f64 a b) < -9.9999999999999996e-95

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 0 90.2%

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

   3. Taylor expanded in t around 0 76.2%

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

   if -9.9999999999999996e-95 < (*.f64 a b) < 2e-281 or 4e-116 < (*.f64 a b) < 5e-32

   1. Initial program 98.6%

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

      1. associate-+l- 98.6%

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

      2. sub-neg 98.6%

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

      3. neg-mul-1 98.6%

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

      4. metadata-eval 98.6%

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

      5. metadata-eval 98.6%

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

      6. cancel-sign-sub-inv 98.6%

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

      7. fma-def 100.0%

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

      8. associate-/l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. *-lft-identity 99.8%

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

      11. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 98.5%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

      4. div-inv 98.6%

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

      5. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in a around inf 79.6%

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

   7. Taylor expanded in a around 0 77.3%

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

   if 2e-281 < (*.f64 a b) < 4e-116 or 5e-32 < (*.f64 a b) < 1e20

   1. Initial program 97.3%

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

   2. Taylor expanded in a around 0 91.5%

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

   3. Taylor expanded in y around 0 83.8%

      \[\leadsto \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+104}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-94}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-281}:\\ \;\;\;\;x \cdot y + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-116}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot y + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;a \cdot b \leq 10^{+20}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 3: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+111} \lor \neg \left(z \cdot t \leq 10^{-33}\right):\\ \;\;\;\;\left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)))
   (if (or (<= (* z t) -1e+111) (not (<= (* z t) 1e-33)))
     (- (+ (* x y) (* (* z t) 0.0625)) t_1)
     (- (+ c (* x y)) t_1))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) * 0.25;
	double tmp;
	if (((z * t) <= -1e+111) || !((z * t) <= 1e-33)) {
		tmp = ((x * y) + ((z * t) * 0.0625)) - t_1;
	} else {
		tmp = (c + (x * y)) - t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) * 0.25d0
    if (((z * t) <= (-1d+111)) .or. (.not. ((z * t) <= 1d-33))) then
        tmp = ((x * y) + ((z * t) * 0.0625d0)) - t_1
    else
        tmp = (c + (x * y)) - t_1
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) * 0.25;
	double tmp;
	if (((z * t) <= -1e+111) || !((z * t) <= 1e-33)) {
		tmp = ((x * y) + ((z * t) * 0.0625)) - t_1;
	} else {
		tmp = (c + (x * y)) - t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	tmp = 0
	if ((z * t) <= -1e+111) or not ((z * t) <= 1e-33):
		tmp = ((x * y) + ((z * t) * 0.0625)) - t_1
	else:
		tmp = (c + (x * y)) - t_1
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) * 0.25)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+111) || !(Float64(z * t) <= 1e-33))
		tmp = Float64(Float64(Float64(x * y) + Float64(Float64(z * t) * 0.0625)) - t_1);
	else
		tmp = Float64(Float64(c + Float64(x * y)) - t_1);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * 0.25;
	tmp = 0.0;
	if (((z * t) <= -1e+111) || ~(((z * t) <= 1e-33)))
		tmp = ((x * y) + ((z * t) * 0.0625)) - t_1;
	else
		tmp = (c + (x * y)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]}, If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+111], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e-33]], $MachinePrecision]], N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999957e110 or 1.0000000000000001e-33 < (*.f64 z t)

   1. Initial program 96.4%

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

      1. associate-+l- 96.4%

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

      2. sub-neg 96.4%

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

      3. neg-mul-1 96.4%

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

      4. metadata-eval 96.4%

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

      5. metadata-eval 96.4%

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

      6. cancel-sign-sub-inv 96.4%

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

      7. fma-def 99.1%

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

      8. associate-/l* 99.0%

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

      9. metadata-eval 99.0%

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

      10. *-lft-identity 99.0%

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

      11. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

      1. fma-udef 96.2%

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

      2. associate-/l* 96.3%

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

      3. +-commutative 96.3%

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

      4. div-inv 96.3%

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

      5. metadata-eval 96.3%

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

   5. Applied egg-rr 96.3%

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

   6. Taylor expanded in a around inf 91.1%

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

   if -9.99999999999999957e110 < (*.f64 z t) < 1.0000000000000001e-33

   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 0 96.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+111} \lor \neg \left(z \cdot t \leq 10^{-33}\right):\\ \;\;\;\;\left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112} \lor \neg \left(a \cdot b \leq 10^{+20}\right):\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* a b) -5e+112) (not (<= (* a b) 1e+20)))
   (- (* x y) (* (* a b) 0.25))
   (+ c (+ (* x y) (* (* z t) 0.0625)))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -5e+112) || !((a * b) <= 1e+20)) {
		tmp = (x * y) - ((a * b) * 0.25);
	} else {
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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) <= (-5d+112)) .or. (.not. ((a * b) <= 1d+20))) then
        tmp = (x * y) - ((a * b) * 0.25d0)
    else
        tmp = c + ((x * y) + ((z * t) * 0.0625d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) <= -5e+112) || !((a * b) <= 1e+20)) {
		tmp = (x * y) - ((a * b) * 0.25);
	} else {
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((a * b) <= -5e+112) or not ((a * b) <= 1e+20):
		tmp = (x * y) - ((a * b) * 0.25)
	else:
		tmp = c + ((x * y) + ((z * t) * 0.0625))
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+112) || !(Float64(a * b) <= 1e+20))
		tmp = 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

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((a * b) <= -5e+112) || ~(((a * b) <= 1e+20)))
		tmp = (x * y) - ((a * b) * 0.25);
	else
		tmp = c + ((x * y) + ((z * t) * 0.0625));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+112], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+20]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $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) < -5e112 or 1e20 < (*.f64 a b)

   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 0 86.7%

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

   3. Taylor expanded in c around 0 82.8%

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

   if -5e112 < (*.f64 a b) < 1e20

   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 a around 0 93.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112} \lor \neg \left(a \cdot b \leq 10^{+20}\right):\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;a \cdot b \leq 10^{+20}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)) (t_2 (* (* z t) 0.0625)))
   (if (<= (* a b) -5e+112)
     (- t_2 t_1)
     (if (<= (* a b) 1e+20) (+ c (+ (* x y) t_2)) (- (* x y) t_1)))))

C

assert(a < b);
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 tmp;
	if ((a * b) <= -5e+112) {
		tmp = t_2 - t_1;
	} else if ((a * b) <= 1e+20) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = (x * y) - t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 = (z * t) * 0.0625d0
    if ((a * b) <= (-5d+112)) then
        tmp = t_2 - t_1
    else if ((a * b) <= 1d+20) then
        tmp = c + ((x * y) + t_2)
    else
        tmp = (x * y) - t_1
    end if
    code = tmp
end function

Java

assert a < b;
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 tmp;
	if ((a * b) <= -5e+112) {
		tmp = t_2 - t_1;
	} else if ((a * b) <= 1e+20) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = (x * y) - t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	t_2 = (z * t) * 0.0625
	tmp = 0
	if (a * b) <= -5e+112:
		tmp = t_2 - t_1
	elif (a * b) <= 1e+20:
		tmp = c + ((x * y) + t_2)
	else:
		tmp = (x * y) - t_1
	return tmp

Julia

a, b = sort([a, b])
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)
	tmp = 0.0
	if (Float64(a * b) <= -5e+112)
		tmp = Float64(t_2 - t_1);
	elseif (Float64(a * b) <= 1e+20)
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	else
		tmp = Float64(Float64(x * y) - t_1);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * 0.25;
	t_2 = (z * t) * 0.0625;
	tmp = 0.0;
	if ((a * b) <= -5e+112)
		tmp = t_2 - t_1;
	elseif ((a * b) <= 1e+20)
		tmp = c + ((x * y) + t_2);
	else
		tmp = (x * y) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
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]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+112], N[(t$95$2 - t$95$1), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+20], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5e112

   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 91.7%

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

   3. Taylor expanded in c around 0 87.7%

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

   if -5e112 < (*.f64 a b) < 1e20

   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 a around 0 93.0%

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

   if 1e20 < (*.f64 a b)

   1. Initial program 95.8%

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

   2. Taylor expanded in z around 0 89.8%

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

   3. Taylor expanded in c around 0 85.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625 - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 10^{+20}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 6: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;t_1 - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(c - \frac{a}{\frac{4}{b}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)))
   (if (<= (* a b) -5e+112)
     (- t_1 (* (* a b) 0.25))
     (if (<= (* a b) 5e+16)
       (+ c (+ (* x y) t_1))
       (+ (* x y) (- c (/ a (/ 4.0 b))))))))

C

assert(a < b);
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) <= -5e+112) {
		tmp = t_1 - ((a * b) * 0.25);
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (x * y) + (c - (a / (4.0 / b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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) <= (-5d+112)) then
        tmp = t_1 - ((a * b) * 0.25d0)
    else if ((a * b) <= 5d+16) then
        tmp = c + ((x * y) + t_1)
    else
        tmp = (x * y) + (c - (a / (4.0d0 / b)))
    end if
    code = tmp
end function

Java

assert a < b;
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) <= -5e+112) {
		tmp = t_1 - ((a * b) * 0.25);
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (x * y) + (c - (a / (4.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (z * t) * 0.0625
	tmp = 0
	if (a * b) <= -5e+112:
		tmp = t_1 - ((a * b) * 0.25)
	elif (a * b) <= 5e+16:
		tmp = c + ((x * y) + t_1)
	else:
		tmp = (x * y) + (c - (a / (4.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	tmp = 0.0
	if (Float64(a * b) <= -5e+112)
		tmp = Float64(t_1 - Float64(Float64(a * b) * 0.25));
	elseif (Float64(a * b) <= 5e+16)
		tmp = Float64(c + Float64(Float64(x * y) + t_1));
	else
		tmp = Float64(Float64(x * y) + Float64(c - Float64(a / Float64(4.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) * 0.0625;
	tmp = 0.0;
	if ((a * b) <= -5e+112)
		tmp = t_1 - ((a * b) * 0.25);
	elseif ((a * b) <= 5e+16)
		tmp = c + ((x * y) + t_1);
	else
		tmp = (x * y) + (c - (a / (4.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+112], N[(t$95$1 - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+16], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(c - N[(a / N[(4.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5e112

   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 91.7%

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

   3. Taylor expanded in c around 0 87.7%

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

   if -5e112 < (*.f64 a b) < 5e16

   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 a around 0 93.7%

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

   if 5e16 < (*.f64 a b)

   1. Initial program 96.0%

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

      1. associate-+l- 96.0%

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

      2. sub-neg 96.0%

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

      3. neg-mul-1 96.0%

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

      4. metadata-eval 96.0%

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

      5. metadata-eval 96.0%

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

      6. cancel-sign-sub-inv 96.0%

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

      7. fma-def 97.9%

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

      8. associate-/l* 97.9%

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

      9. metadata-eval 97.9%

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

      10. *-lft-identity 97.9%

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

      11. associate-/l* 97.8%

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

   3. Simplified 97.8%

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

      1. fma-udef 95.8%

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

      2. associate-/l* 95.8%

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

      3. +-commutative 95.8%

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

      4. div-inv 95.8%

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

      5. metadata-eval 95.8%

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

   5. Applied egg-rr 95.8%

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

   6. Taylor expanded in z around 0 88.3%

      \[\leadsto \color{blue}{y \cdot x} - \left(\frac{a}{\frac{4}{b}} - c\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625 - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(c - \frac{a}{\frac{4}{b}}\right)\\ \end{array} \]

Alternative 7: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* a b) 0.25)) (t_2 (* (* z t) 0.0625)))
   (if (<= (* a b) -5e+112)
     (- t_2 t_1)
     (if (<= (* a b) 5e+16) (+ c (+ (* x y) t_2)) (- (+ c (* x y)) t_1)))))

C

assert(a < b);
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 tmp;
	if ((a * b) <= -5e+112) {
		tmp = t_2 - t_1;
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = (c + (x * y)) - t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 = (z * t) * 0.0625d0
    if ((a * b) <= (-5d+112)) then
        tmp = t_2 - t_1
    else if ((a * b) <= 5d+16) then
        tmp = c + ((x * y) + t_2)
    else
        tmp = (c + (x * y)) - t_1
    end if
    code = tmp
end function

Java

assert a < b;
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 tmp;
	if ((a * b) <= -5e+112) {
		tmp = t_2 - t_1;
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_2);
	} else {
		tmp = (c + (x * y)) - t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (a * b) * 0.25
	t_2 = (z * t) * 0.0625
	tmp = 0
	if (a * b) <= -5e+112:
		tmp = t_2 - t_1
	elif (a * b) <= 5e+16:
		tmp = c + ((x * y) + t_2)
	else:
		tmp = (c + (x * y)) - t_1
	return tmp

Julia

a, b = sort([a, b])
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)
	tmp = 0.0
	if (Float64(a * b) <= -5e+112)
		tmp = Float64(t_2 - t_1);
	elseif (Float64(a * b) <= 5e+16)
		tmp = Float64(c + Float64(Float64(x * y) + t_2));
	else
		tmp = Float64(Float64(c + Float64(x * y)) - t_1);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) * 0.25;
	t_2 = (z * t) * 0.0625;
	tmp = 0.0;
	if ((a * b) <= -5e+112)
		tmp = t_2 - t_1;
	elseif ((a * b) <= 5e+16)
		tmp = c + ((x * y) + t_2);
	else
		tmp = (c + (x * y)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
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]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+112], N[(t$95$2 - t$95$1), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+16], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5e112

   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 91.7%

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

   3. Taylor expanded in c around 0 87.7%

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

   if -5e112 < (*.f64 a b) < 5e16

   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 a around 0 93.7%

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

   if 5e16 < (*.f64 a b)

   1. Initial program 96.0%

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

   2. Taylor expanded in z around 0 88.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625 - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 8: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ t_2 := \left(a \cdot b\right) \cdot 0.25\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(c + t_1\right) - t_2\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* z t) 0.0625)) (t_2 (* (* a b) 0.25)))
   (if (<= (* a b) -5e+112)
     (- (+ c t_1) t_2)
     (if (<= (* a b) 5e+16) (+ c (+ (* x y) t_1)) (- (+ c (* x y)) t_2)))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) * 0.0625;
	double t_2 = (a * b) * 0.25;
	double tmp;
	if ((a * b) <= -5e+112) {
		tmp = (c + t_1) - t_2;
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (c + (x * y)) - t_2;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 = (z * t) * 0.0625d0
    t_2 = (a * b) * 0.25d0
    if ((a * b) <= (-5d+112)) then
        tmp = (c + t_1) - t_2
    else if ((a * b) <= 5d+16) then
        tmp = c + ((x * y) + t_1)
    else
        tmp = (c + (x * y)) - t_2
    end if
    code = tmp
end function

Java

assert a < b;
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 t_2 = (a * b) * 0.25;
	double tmp;
	if ((a * b) <= -5e+112) {
		tmp = (c + t_1) - t_2;
	} else if ((a * b) <= 5e+16) {
		tmp = c + ((x * y) + t_1);
	} else {
		tmp = (c + (x * y)) - t_2;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = (z * t) * 0.0625
	t_2 = (a * b) * 0.25
	tmp = 0
	if (a * b) <= -5e+112:
		tmp = (c + t_1) - t_2
	elif (a * b) <= 5e+16:
		tmp = c + ((x * y) + t_1)
	else:
		tmp = (c + (x * y)) - t_2
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) * 0.0625)
	t_2 = Float64(Float64(a * b) * 0.25)
	tmp = 0.0
	if (Float64(a * b) <= -5e+112)
		tmp = Float64(Float64(c + t_1) - t_2);
	elseif (Float64(a * b) <= 5e+16)
		tmp = Float64(c + Float64(Float64(x * y) + t_1));
	else
		tmp = Float64(Float64(c + Float64(x * y)) - t_2);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) * 0.0625;
	t_2 = (a * b) * 0.25;
	tmp = 0.0;
	if ((a * b) <= -5e+112)
		tmp = (c + t_1) - t_2;
	elseif ((a * b) <= 5e+16)
		tmp = c + ((x * y) + t_1);
	else
		tmp = (c + (x * y)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+112], N[(N[(c + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+16], N[(c + N[(N[(x * y), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5e112

   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 91.7%

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

   if -5e112 < (*.f64 a b) < 5e16

   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 a around 0 93.7%

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

   if 5e16 < (*.f64 a b)

   1. Initial program 96.0%

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

   2. Taylor expanded in z around 0 88.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(c + \left(z \cdot t\right) \cdot 0.0625\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+16}:\\ \;\;\;\;c + \left(x \cdot y + \left(z \cdot t\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 9: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := c + x \cdot y\\ t_2 := c + \left(z \cdot t\right) \cdot 0.0625\\ t_3 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-248}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* x y)))
        (t_2 (+ c (* (* z t) 0.0625)))
        (t_3 (* a (* b -0.25))))
   (if (<= z -1.25e+123)
     t_2
     (if (<= z -1.95e-204)
       t_1
       (if (<= z -4.2e-248)
         t_3
         (if (<= z 9.4e-154) t_1 (if (<= z 5.7e-18) t_3 t_2)))))))

C

assert(a < b);
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 = c + ((z * t) * 0.0625);
	double t_3 = a * (b * -0.25);
	double tmp;
	if (z <= -1.25e+123) {
		tmp = t_2;
	} else if (z <= -1.95e-204) {
		tmp = t_1;
	} else if (z <= -4.2e-248) {
		tmp = t_3;
	} else if (z <= 9.4e-154) {
		tmp = t_1;
	} else if (z <= 5.7e-18) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = c + (x * y)
    t_2 = c + ((z * t) * 0.0625d0)
    t_3 = a * (b * (-0.25d0))
    if (z <= (-1.25d+123)) then
        tmp = t_2
    else if (z <= (-1.95d-204)) then
        tmp = t_1
    else if (z <= (-4.2d-248)) then
        tmp = t_3
    else if (z <= 9.4d-154) then
        tmp = t_1
    else if (z <= 5.7d-18) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert a < b;
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 = c + ((z * t) * 0.0625);
	double t_3 = a * (b * -0.25);
	double tmp;
	if (z <= -1.25e+123) {
		tmp = t_2;
	} else if (z <= -1.95e-204) {
		tmp = t_1;
	} else if (z <= -4.2e-248) {
		tmp = t_3;
	} else if (z <= 9.4e-154) {
		tmp = t_1;
	} else if (z <= 5.7e-18) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = c + (x * y)
	t_2 = c + ((z * t) * 0.0625)
	t_3 = a * (b * -0.25)
	tmp = 0
	if z <= -1.25e+123:
		tmp = t_2
	elif z <= -1.95e-204:
		tmp = t_1
	elif z <= -4.2e-248:
		tmp = t_3
	elif z <= 9.4e-154:
		tmp = t_1
	elif z <= 5.7e-18:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(x * y))
	t_2 = Float64(c + Float64(Float64(z * t) * 0.0625))
	t_3 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (z <= -1.25e+123)
		tmp = t_2;
	elseif (z <= -1.95e-204)
		tmp = t_1;
	elseif (z <= -4.2e-248)
		tmp = t_3;
	elseif (z <= 9.4e-154)
		tmp = t_1;
	elseif (z <= 5.7e-18)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + (x * y);
	t_2 = c + ((z * t) * 0.0625);
	t_3 = a * (b * -0.25);
	tmp = 0.0;
	if (z <= -1.25e+123)
		tmp = t_2;
	elseif (z <= -1.95e-204)
		tmp = t_1;
	elseif (z <= -4.2e-248)
		tmp = t_3;
	elseif (z <= 9.4e-154)
		tmp = t_1;
	elseif (z <= 5.7e-18)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e+123], t$95$2, If[LessEqual[z, -1.95e-204], t$95$1, If[LessEqual[z, -4.2e-248], t$95$3, If[LessEqual[z, 9.4e-154], t$95$1, If[LessEqual[z, 5.7e-18], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999994e123 or 5.69999999999999971e-18 < z

   1. Initial program 97.1%

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

   2. Taylor expanded in a around 0 79.4%

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

   3. Taylor expanded in y around 0 64.1%

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

   if -1.24999999999999994e123 < z < -1.95e-204 or -4.2e-248 < z < 9.4000000000000003e-154

   1. Initial program 98.2%

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

   2. Taylor expanded in a around 0 68.0%

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

   3. Taylor expanded in t around 0 55.8%

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

   if -1.95e-204 < z < -4.2e-248 or 9.4000000000000003e-154 < z < 5.69999999999999971e-18

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. fma-def 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a around inf 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*l* 54.8%

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

   8. Simplified 54.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-204}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-248}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-154}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-18}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \end{array} \]

Alternative 10: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+158} \lor \neg \left(z \leq -1.12 \cdot 10^{+104} \lor \neg \left(z \leq -1.9 \cdot 10^{-8}\right) \land z \leq 2.7 \cdot 10^{-110}\right):\\ \;\;\;\;x \cdot y + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= z -4.4e+158)
         (not
          (or (<= z -1.12e+104) (and (not (<= z -1.9e-8)) (<= z 2.7e-110)))))
   (+ (* x y) (* (* z t) 0.0625))
   (+ c (* b (* a -0.25)))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z <= -4.4e+158) || !((z <= -1.12e+104) || (!(z <= -1.9e-8) && (z <= 2.7e-110)))) {
		tmp = (x * y) + ((z * t) * 0.0625);
	} else {
		tmp = c + (b * (a * -0.25));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 ((z <= (-4.4d+158)) .or. (.not. (z <= (-1.12d+104)) .or. (.not. (z <= (-1.9d-8))) .and. (z <= 2.7d-110))) then
        tmp = (x * y) + ((z * t) * 0.0625d0)
    else
        tmp = c + (b * (a * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z <= -4.4e+158) || !((z <= -1.12e+104) || (!(z <= -1.9e-8) && (z <= 2.7e-110)))) {
		tmp = (x * y) + ((z * t) * 0.0625);
	} else {
		tmp = c + (b * (a * -0.25));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if (z <= -4.4e+158) or not ((z <= -1.12e+104) or (not (z <= -1.9e-8) and (z <= 2.7e-110))):
		tmp = (x * y) + ((z * t) * 0.0625)
	else:
		tmp = c + (b * (a * -0.25))
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((z <= -4.4e+158) || !((z <= -1.12e+104) || (!(z <= -1.9e-8) && (z <= 2.7e-110))))
		tmp = Float64(Float64(x * y) + Float64(Float64(z * t) * 0.0625));
	else
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((z <= -4.4e+158) || ~(((z <= -1.12e+104) || (~((z <= -1.9e-8)) && (z <= 2.7e-110)))))
		tmp = (x * y) + ((z * t) * 0.0625);
	else
		tmp = c + (b * (a * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[z, -4.4e+158], N[Not[Or[LessEqual[z, -1.12e+104], And[N[Not[LessEqual[z, -1.9e-8]], $MachinePrecision], LessEqual[z, 2.7e-110]]]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000002e158 or -1.12000000000000003e104 < z < -1.90000000000000014e-8 or 2.6999999999999998e-110 < z

   1. Initial program 97.9%

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

      1. associate-+l- 97.9%

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

      2. sub-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. metadata-eval 97.9%

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

      5. metadata-eval 97.9%

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

      6. cancel-sign-sub-inv 97.9%

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

      7. fma-def 100.0%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 97.7%

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

      2. associate-/l* 97.8%

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

      3. +-commutative 97.8%

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

      4. div-inv 97.8%

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

      5. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in a around inf 86.5%

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

   7. Taylor expanded in a around 0 63.9%

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

   if -4.4000000000000002e158 < z < -1.12000000000000003e104 or -1.90000000000000014e-8 < z < 2.6999999999999998e-110

   1. Initial program 98.2%

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

   2. Taylor expanded in a around inf 63.4%

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

      1. *-commutative 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

   4. Simplified 63.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+158} \lor \neg \left(z \leq -1.12 \cdot 10^{+104} \lor \neg \left(z \leq -1.9 \cdot 10^{-8}\right) \land z \leq 2.7 \cdot 10^{-110}\right):\\ \;\;\;\;x \cdot y + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \end{array} \]

Alternative 11: 97.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (+ c (- (+ (/ (* z t) 16.0) (* x y)) (/ (* a b) 4.0))))

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 + ((((z * t) / 16.0d0) + (x * y)) - ((a * b) / 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(x, y, z, t, a, b, c)
	tmp = c + ((((z * t) / 16.0) + (x * y)) - ((a * b) / 4.0));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := N[(c + N[(N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

2. Final simplification 98.0%

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

Alternative 12: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+179} \lor \neg \left(a \leq -1.05 \cdot 10^{+141}\right) \land \left(a \leq -4.1 \cdot 10^{+89} \lor \neg \left(a \leq 2.9 \cdot 10^{-49}\right)\right):\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= a -4.8e+179)
         (and (not (<= a -1.05e+141))
              (or (<= a -4.1e+89) (not (<= a 2.9e-49)))))
   (* a (* b -0.25))
   (+ c (* x y))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a <= -4.8e+179) || (!(a <= -1.05e+141) && ((a <= -4.1e+89) || !(a <= 2.9e-49)))) {
		tmp = a * (b * -0.25);
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 <= (-4.8d+179)) .or. (.not. (a <= (-1.05d+141))) .and. (a <= (-4.1d+89)) .or. (.not. (a <= 2.9d-49))) then
        tmp = a * (b * (-0.25d0))
    else
        tmp = c + (x * y)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a <= -4.8e+179) || (!(a <= -1.05e+141) && ((a <= -4.1e+89) || !(a <= 2.9e-49)))) {
		tmp = a * (b * -0.25);
	} else {
		tmp = c + (x * y);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if (a <= -4.8e+179) or (not (a <= -1.05e+141) and ((a <= -4.1e+89) or not (a <= 2.9e-49))):
		tmp = a * (b * -0.25)
	else:
		tmp = c + (x * y)
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((a <= -4.8e+179) || (!(a <= -1.05e+141) && ((a <= -4.1e+89) || !(a <= 2.9e-49))))
		tmp = Float64(a * Float64(b * -0.25));
	else
		tmp = Float64(c + Float64(x * y));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((a <= -4.8e+179) || (~((a <= -1.05e+141)) && ((a <= -4.1e+89) || ~((a <= 2.9e-49)))))
		tmp = a * (b * -0.25);
	else
		tmp = c + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[a, -4.8e+179], And[N[Not[LessEqual[a, -1.05e+141]], $MachinePrecision], Or[LessEqual[a, -4.1e+89], N[Not[LessEqual[a, 2.9e-49]], $MachinePrecision]]]], N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.80000000000000025e179 or -1.0499999999999999e141 < a < -4.09999999999999985e89 or 2.9e-49 < a

   1. Initial program 97.5%

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

      1. associate-+l- 97.5%

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

      2. sub-neg 97.5%

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

      3. neg-mul-1 97.5%

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

      4. metadata-eval 97.5%

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

      5. metadata-eval 97.5%

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

      6. cancel-sign-sub-inv 97.5%

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

      7. fma-def 99.2%

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

      8. associate-/l* 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-/l* 99.0%

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

   3. Simplified 99.0%

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

      1. fma-udef 97.4%

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

      2. associate-/l* 97.5%

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

      3. +-commutative 97.5%

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

      4. div-inv 97.5%

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

      5. metadata-eval 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in a around inf 51.4%

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

      1. *-commutative 51.4%

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

      2. associate-*l* 51.4%

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

   8. Simplified 51.4%

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

   if -4.80000000000000025e179 < a < -1.0499999999999999e141 or -4.09999999999999985e89 < a < 2.9e-49

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0 87.1%

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

   3. Taylor expanded in t around 0 56.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+179} \lor \neg \left(a \leq -1.05 \cdot 10^{+141}\right) \land \left(a \leq -4.1 \cdot 10^{+89} \lor \neg \left(a \leq 2.9 \cdot 10^{-49}\right)\right):\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 13: 40.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;b \leq -3.35 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-191}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq -6.3 \cdot 10^{-293}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= b -3.35e-31)
     t_1
     (if (<= b -6e-191)
       (* x y)
       (if (<= b -6.3e-293) c (if (<= b 2.2e+39) (* x y) t_1))))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -3.35e-31) {
		tmp = t_1;
	} else if (b <= -6e-191) {
		tmp = x * y;
	} else if (b <= -6.3e-293) {
		tmp = c;
	} else if (b <= 2.2e+39) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * (-0.25d0))
    if (b <= (-3.35d-31)) then
        tmp = t_1
    else if (b <= (-6d-191)) then
        tmp = x * y
    else if (b <= (-6.3d-293)) then
        tmp = c
    else if (b <= 2.2d+39) then
        tmp = x * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = a * (b * -0.25);
	double tmp;
	if (b <= -3.35e-31) {
		tmp = t_1;
	} else if (b <= -6e-191) {
		tmp = x * y;
	} else if (b <= -6.3e-293) {
		tmp = c;
	} else if (b <= 2.2e+39) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = a * (b * -0.25)
	tmp = 0
	if b <= -3.35e-31:
		tmp = t_1
	elif b <= -6e-191:
		tmp = x * y
	elif b <= -6.3e-293:
		tmp = c
	elif b <= 2.2e+39:
		tmp = x * y
	else:
		tmp = t_1
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (b <= -3.35e-31)
		tmp = t_1;
	elseif (b <= -6e-191)
		tmp = Float64(x * y);
	elseif (b <= -6.3e-293)
		tmp = c;
	elseif (b <= 2.2e+39)
		tmp = Float64(x * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = a * (b * -0.25);
	tmp = 0.0;
	if (b <= -3.35e-31)
		tmp = t_1;
	elseif (b <= -6e-191)
		tmp = x * y;
	elseif (b <= -6.3e-293)
		tmp = c;
	elseif (b <= 2.2e+39)
		tmp = x * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.35e-31], t$95$1, If[LessEqual[b, -6e-191], N[(x * y), $MachinePrecision], If[LessEqual[b, -6.3e-293], c, If[LessEqual[b, 2.2e+39], N[(x * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.35000000000000002e-31 or 2.2000000000000001e39 < b

   1. Initial program 96.9%

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

      1. associate-+l- 96.9%

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

      2. sub-neg 96.9%

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

      3. neg-mul-1 96.9%

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

      4. metadata-eval 96.9%

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

      5. metadata-eval 96.9%

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

      6. cancel-sign-sub-inv 96.9%

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

      7. fma-def 99.2%

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

      8. associate-/l* 99.2%

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

      9. metadata-eval 99.2%

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

      10. *-lft-identity 99.2%

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

      11. associate-/l* 99.0%

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

   3. Simplified 99.0%

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

      1. fma-udef 96.7%

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

      2. associate-/l* 96.7%

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

      3. +-commutative 96.7%

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

      4. div-inv 96.7%

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

      5. metadata-eval 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in a around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 51.3%

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

   8. Simplified 51.3%

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

   if -3.35000000000000002e-31 < b < -6.0000000000000001e-191 or -6.29999999999999988e-293 < b < 2.2000000000000001e39

   1. Initial program 99.0%

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

      1. associate-+l- 99.0%

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

      2. sub-neg 99.0%

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

      3. neg-mul-1 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. cancel-sign-sub-inv 99.0%

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

      7. fma-def 99.0%

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

      8. associate-/l* 98.9%

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

      9. metadata-eval 98.9%

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

      10. *-lft-identity 98.9%

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

      11. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

      1. fma-udef 98.9%

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

      2. associate-/l* 99.0%

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

      3. +-commutative 99.0%

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

      4. div-inv 99.0%

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

      5. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in a around inf 82.7%

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

   7. Taylor expanded in x around inf 44.4%

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

   if -6.0000000000000001e-191 < b < -6.29999999999999988e-293

   1. Initial program 99.9%

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

   2. Taylor expanded in c around inf 46.2%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.35 \cdot 10^{-31}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-191}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq -6.3 \cdot 10^{-293}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \end{array} \]

Alternative 14: 38.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-86}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;c\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* t (* z 0.0625))))
   (if (<= z -6.5e+122)
     t_1
     (if (<= z -2.4e-86)
       (* x y)
       (if (<= z -2.8e-203) c (if (<= z 4.5e-13) (* a (* b -0.25)) t_1))))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = t * (z * 0.0625);
	double tmp;
	if (z <= -6.5e+122) {
		tmp = t_1;
	} else if (z <= -2.4e-86) {
		tmp = x * y;
	} else if (z <= -2.8e-203) {
		tmp = c;
	} else if (z <= 4.5e-13) {
		tmp = a * (b * -0.25);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 = t * (z * 0.0625d0)
    if (z <= (-6.5d+122)) then
        tmp = t_1
    else if (z <= (-2.4d-86)) then
        tmp = x * y
    else if (z <= (-2.8d-203)) then
        tmp = c
    else if (z <= 4.5d-13) then
        tmp = a * (b * (-0.25d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = t * (z * 0.0625);
	double tmp;
	if (z <= -6.5e+122) {
		tmp = t_1;
	} else if (z <= -2.4e-86) {
		tmp = x * y;
	} else if (z <= -2.8e-203) {
		tmp = c;
	} else if (z <= 4.5e-13) {
		tmp = a * (b * -0.25);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = t * (z * 0.0625)
	tmp = 0
	if z <= -6.5e+122:
		tmp = t_1
	elif z <= -2.4e-86:
		tmp = x * y
	elif z <= -2.8e-203:
		tmp = c
	elif z <= 4.5e-13:
		tmp = a * (b * -0.25)
	else:
		tmp = t_1
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(t * Float64(z * 0.0625))
	tmp = 0.0
	if (z <= -6.5e+122)
		tmp = t_1;
	elseif (z <= -2.4e-86)
		tmp = Float64(x * y);
	elseif (z <= -2.8e-203)
		tmp = c;
	elseif (z <= 4.5e-13)
		tmp = Float64(a * Float64(b * -0.25));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = t * (z * 0.0625);
	tmp = 0.0;
	if (z <= -6.5e+122)
		tmp = t_1;
	elseif (z <= -2.4e-86)
		tmp = x * y;
	elseif (z <= -2.8e-203)
		tmp = c;
	elseif (z <= 4.5e-13)
		tmp = a * (b * -0.25);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+122], t$95$1, If[LessEqual[z, -2.4e-86], N[(x * y), $MachinePrecision], If[LessEqual[z, -2.8e-203], c, If[LessEqual[z, 4.5e-13], N[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.49999999999999963e122 or 4.5e-13 < z

   1. Initial program 97.0%

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

      1. associate-+l- 97.0%

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

      2. sub-neg 97.0%

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

      3. neg-mul-1 97.0%

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

      4. metadata-eval 97.0%

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

      5. metadata-eval 97.0%

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

      6. cancel-sign-sub-inv 97.0%

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

      7. fma-def 100.0%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 96.9%

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

      2. associate-/l* 97.0%

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

      3. +-commutative 97.0%

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

      4. div-inv 97.0%

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

      5. metadata-eval 97.0%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in a around inf 85.5%

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

   7. Taylor expanded in z around inf 52.9%

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

      1. associate-*r* 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-*l* 52.9%

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

   9. Simplified 52.9%

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

   if -6.49999999999999963e122 < z < -2.40000000000000013e-86

   1. Initial program 98.0%

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

      1. associate-+l- 98.0%

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

      2. sub-neg 98.0%

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

      3. neg-mul-1 98.0%

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

      4. metadata-eval 98.0%

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

      5. metadata-eval 98.0%

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

      6. cancel-sign-sub-inv 98.0%

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

      7. fma-def 98.0%

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

      8. associate-/l* 98.0%

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

      9. metadata-eval 98.0%

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

      10. *-lft-identity 98.0%

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

      11. associate-/l* 97.9%

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

   3. Simplified 97.9%

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

      1. fma-udef 97.9%

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

      2. associate-/l* 98.0%

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

      3. +-commutative 98.0%

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

      4. div-inv 98.0%

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

      5. metadata-eval 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in a around inf 82.5%

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

   7. Taylor expanded in x around inf 38.5%

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

   if -2.40000000000000013e-86 < z < -2.80000000000000022e-203

   1. Initial program 99.8%

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

   2. Taylor expanded in c around inf 41.7%

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

   if -2.80000000000000022e-203 < z < 4.5e-13

   1. Initial program 98.8%

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

      1. associate-+l- 98.8%

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

      2. sub-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. metadata-eval 98.8%

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

      5. metadata-eval 98.8%

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

      6. cancel-sign-sub-inv 98.8%

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

      7. fma-def 98.8%

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

      8. associate-/l* 98.8%

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

      9. metadata-eval 98.8%

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

      10. *-lft-identity 98.8%

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

      11. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

      1. fma-udef 98.6%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

      4. div-inv 98.6%

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

      5. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in a around inf 47.5%

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

      1. *-commutative 47.5%

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

      2. associate-*l* 47.5%

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

   8. Simplified 47.5%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -0.25\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+122}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-86}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;c\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot 0.0625\right)\\ \end{array} \]

Alternative 15: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_1 := c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-8}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ c (* (* z t) 0.0625))))
   (if (<= z -6.6e+122)
     t_1
     (if (<= z -7.2e-8)
       (+ c (* x y))
       (if (<= z 9.8e-8) (+ c (* b (* a -0.25))) t_1)))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + ((z * t) * 0.0625);
	double tmp;
	if (z <= -6.6e+122) {
		tmp = t_1;
	} else if (z <= -7.2e-8) {
		tmp = c + (x * y);
	} else if (z <= 9.8e-8) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 + ((z * t) * 0.0625d0)
    if (z <= (-6.6d+122)) then
        tmp = t_1
    else if (z <= (-7.2d-8)) then
        tmp = c + (x * y)
    else if (z <= 9.8d-8) then
        tmp = c + (b * (a * (-0.25d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = c + ((z * t) * 0.0625);
	double tmp;
	if (z <= -6.6e+122) {
		tmp = t_1;
	} else if (z <= -7.2e-8) {
		tmp = c + (x * y);
	} else if (z <= 9.8e-8) {
		tmp = c + (b * (a * -0.25));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	t_1 = c + ((z * t) * 0.0625)
	tmp = 0
	if z <= -6.6e+122:
		tmp = t_1
	elif z <= -7.2e-8:
		tmp = c + (x * y)
	elif z <= 9.8e-8:
		tmp = c + (b * (a * -0.25))
	else:
		tmp = t_1
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(c + Float64(Float64(z * t) * 0.0625))
	tmp = 0.0
	if (z <= -6.6e+122)
		tmp = t_1;
	elseif (z <= -7.2e-8)
		tmp = Float64(c + Float64(x * y));
	elseif (z <= 9.8e-8)
		tmp = Float64(c + Float64(b * Float64(a * -0.25)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = c + ((z * t) * 0.0625);
	tmp = 0.0;
	if (z <= -6.6e+122)
		tmp = t_1;
	elseif (z <= -7.2e-8)
		tmp = c + (x * y);
	elseif (z <= 9.8e-8)
		tmp = c + (b * (a * -0.25));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(c + N[(N[(z * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+122], t$95$1, If[LessEqual[z, -7.2e-8], N[(c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.8e-8], N[(c + N[(b * N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5999999999999998e122 or 9.8000000000000004e-8 < z

   1. Initial program 97.0%

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

   2. Taylor expanded in a around 0 79.6%

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

   3. Taylor expanded in y around 0 64.0%

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

   if -6.5999999999999998e122 < z < -7.19999999999999962e-8

   1. Initial program 96.9%

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

   2. Taylor expanded in a around 0 75.3%

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

   3. Taylor expanded in t around 0 53.2%

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

   if -7.19999999999999962e-8 < z < 9.8000000000000004e-8

   1. Initial program 99.1%

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

   2. Taylor expanded in a around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-*r* 65.3%

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

   4. Simplified 65.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+122}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-8}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \end{array} \]

Alternative 16: 35.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+92}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-79}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= x -2.3e+92) (* x y) (if (<= x 2.3e-79) c (* x y))))

C

assert(a < b);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (x <= -2.3e+92) {
		tmp = x * y;
	} else if (x <= 2.3e-79) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 <= (-2.3d+92)) then
        tmp = x * y
    else if (x <= 2.3d-79) then
        tmp = c
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (x <= -2.3e+92) {
		tmp = x * y;
	} else if (x <= 2.3e-79) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if x <= -2.3e+92:
		tmp = x * y
	elif x <= 2.3e-79:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (x <= -2.3e+92)
		tmp = Float64(x * y);
	elseif (x <= 2.3e-79)
		tmp = c;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (x <= -2.3e+92)
		tmp = x * y;
	elseif (x <= 2.3e-79)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[x, -2.3e+92], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.3e-79], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999998e92 or 2.30000000000000012e-79 < x

   1. Initial program 96.2%

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

      1. associate-+l- 96.2%

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

      2. sub-neg 96.2%

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

      3. neg-mul-1 96.2%

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

      4. metadata-eval 96.2%

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

      5. metadata-eval 96.2%

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

      6. cancel-sign-sub-inv 96.2%

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

      7. fma-def 98.4%

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

      8. associate-/l* 98.4%

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

      9. metadata-eval 98.4%

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

      10. *-lft-identity 98.4%

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

      11. associate-/l* 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 96.1%

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

      2. associate-/l* 96.1%

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

      3. +-commutative 96.1%

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

      4. div-inv 96.1%

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

      5. metadata-eval 96.1%

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

   5. Applied egg-rr 96.1%

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

   6. Taylor expanded in a around inf 82.9%

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

   7. Taylor expanded in x around inf 42.3%

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

   if -2.29999999999999998e92 < x < 2.30000000000000012e-79

   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 25.1%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+92}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-79}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17: 22.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ c \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c) :precision binary64 c)

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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

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

Python

[a, b] = sort([a, b])
def code(x, y, z, t, a, b, c):
	return c

Julia

a, b = sort([a, b])
function code(x, y, z, t, a, b, c)
	return c
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(x, y, z, t, a, b, c)
	tmp = c;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

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 c around inf 20.0%

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

3. Final simplification 20.0%

   \[\leadsto c \]

Reproduce

herbie shell --seed 2023221 
(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))