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

Percentage Accurate: 97.9% → 98.5%

Time: 11.0s

Alternatives: 12

Speedup: 1.6×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (fma a (* b -0.25) (fma 0.0625 (* t z) (fma x y c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(a, (b * -0.25), fma(0.0625, (t * z), fma(x, y, c)));
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(a, Float64(b * -0.25), fma(0.0625, Float64(t * z), fma(x, y, c)))
end

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. associate-+r+ N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. +-commutative N/A

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

   16. accelerator-lowering-fma.f64 99.6

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, t \cdot \left(0.0625 \cdot z\right)\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma x y (* t (* 0.0625 z)))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -5e+154) t_1 (if (<= t_2 1e+175) (fma (* b -0.25) a c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(x, y, (t * (0.0625 * z)));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -5e+154) {
		tmp = t_1;
	} else if (t_2 <= 1e+175) {
		tmp = fma((b * -0.25), a, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(x, y, Float64(t * Float64(0.0625 * z)))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -5e+154)
		tmp = t_1;
	elseif (t_2 <= 1e+175)
		tmp = fma(Float64(b * -0.25), a, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x * y + N[(t * N[(0.0625 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+154], t$95$1, If[LessEqual[t$95$2, 1e+175], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.00000000000000004e154 or 9.9999999999999994e174 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 88.3

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

   8. Simplified 88.3%

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

   if -5.00000000000000004e154 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 9.9999999999999994e174

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 72.6

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 72.6

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

   7. Applied egg-rr 72.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot \left(0.0625 \cdot z\right)\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot \left(0.0625 \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{if}\;t \cdot z \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 z) t c)))
   (if (<= (* t z) -1.5e+115)
     t_1
     (if (<= (* t z) -1e+17)
       (fma y x c)
       (if (<= (* t z) -5e-311)
         (fma (* b -0.25) a c)
         (if (<= (* t z) 1e+106) (fma y x c) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * z), t, c);
	double tmp;
	if ((t * z) <= -1.5e+115) {
		tmp = t_1;
	} else if ((t * z) <= -1e+17) {
		tmp = fma(y, x, c);
	} else if ((t * z) <= -5e-311) {
		tmp = fma((b * -0.25), a, c);
	} else if ((t * z) <= 1e+106) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * z), t, c)
	tmp = 0.0
	if (Float64(t * z) <= -1.5e+115)
		tmp = t_1;
	elseif (Float64(t * z) <= -1e+17)
		tmp = fma(y, x, c);
	elseif (Float64(t * z) <= -5e-311)
		tmp = fma(Float64(b * -0.25), a, c);
	elseif (Float64(t * z) <= 1e+106)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.5e+115], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], -1e+17], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], -5e-311], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+106], N[(y * x + c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.5e115 or 1.00000000000000009e106 < (*.f64 z t)

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]

      2. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{16} \cdot t\right)} + c \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(z \cdot \frac{1}{16}\right) \cdot t} + c \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \frac{1}{16}, t, c\right)} \]

      5. *-lowering-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -1.5e115 < (*.f64 z t) < -1e17 or -5.00000000000023e-311 < (*.f64 z t) < 1.00000000000000009e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 67.6

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

   5. Simplified 67.6%

      \[\leadsto \color{blue}{x \cdot y} + c \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 67.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]

   if -1e17 < (*.f64 z t) < -5.00000000000023e-311

   1. Initial program 97.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 82.2

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

   5. Simplified 82.2%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 82.2

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

   7. Applied egg-rr 82.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{elif}\;t \cdot z \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1e+46)
   (fma 0.0625 (* t z) (fma x y c))
   (if (<= (* x y) 4e+114)
     (fma 0.0625 (* t z) (fma a (* b -0.25) c))
     (fma a (* b -0.25) (fma x y c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1e+46) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else if ((x * y) <= 4e+114) {
		tmp = fma(0.0625, (t * z), fma(a, (b * -0.25), c));
	} else {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1e+46)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	elseif (Float64(x * y) <= 4e+114)
		tmp = fma(0.0625, Float64(t * z), fma(a, Float64(b * -0.25), c));
	else
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+46], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+114], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999999e45

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.0

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

   5. Simplified 93.0%

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

   if -9.9999999999999999e45 < (*.f64 x y) < 4e114

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 94.3

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

   5. Simplified 94.3%

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

   if 4e114 < (*.f64 x y)

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 91.4

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

   5. Simplified 91.4%

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

Alternative 5: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -1e+162)
     t_1
     (if (<= (* t z) -5e-311)
       (fma (* b -0.25) a c)
       (if (<= (* t z) 1e+106) (fma y x c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -1e+162) {
		tmp = t_1;
	} else if ((t * z) <= -5e-311) {
		tmp = fma((b * -0.25), a, c);
	} else if ((t * z) <= 1e+106) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1e+162)
		tmp = t_1;
	elseif (Float64(t * z) <= -5e-311)
		tmp = fma(Float64(b * -0.25), a, c);
	elseif (Float64(t * z) <= 1e+106)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1e+162], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], -5e-311], N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+106], N[(y * x + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999994e161 or 1.00000000000000009e106 < (*.f64 z t)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]

      2. *-lowering-*.f64 76.9

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

   5. Simplified 76.9%

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

   if -9.9999999999999994e161 < (*.f64 z t) < -5.00000000000023e-311

   1. Initial program 98.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 68.1

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

   5. Simplified 68.1%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 68.1

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

   7. Applied egg-rr 68.1%

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

   if -5.00000000000023e-311 < (*.f64 z t) < 1.00000000000000009e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 66.6

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

   5. Simplified 66.6%

      \[\leadsto \color{blue}{x \cdot y} + c \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 66.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]

   7. Applied egg-rr 66.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+162}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot -0.25, a, c\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* a b) -2e+134)
   (fma a (* b -0.25) (fma x y c))
   (if (<= (* a b) 2e+123)
     (fma 0.0625 (* t z) (fma x y c))
     (fma a (* b -0.25) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a * b) <= -2e+134) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else if ((a * b) <= 2e+123) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = fma(a, (b * -0.25), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(a * b) <= -2e+134)
		tmp = fma(a, Float64(b * -0.25), fma(x, y, c));
	elseif (Float64(a * b) <= 2e+123)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = fma(a, Float64(b * -0.25), Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+134], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+123], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.99999999999999984e134

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 92.1

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

   5. Simplified 92.1%

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

   if -1.99999999999999984e134 < (*.f64 a b) < 1.99999999999999996e123

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.3

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

   5. Simplified 93.3%

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

   if 1.99999999999999996e123 < (*.f64 a b)

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 91.0

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

   8. Simplified 91.0%

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

Alternative 7: 88.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (* x y))))
   (if (<= (* a b) -2e+134)
     t_1
     (if (<= (* a b) 2e+123) (fma 0.0625 (* t z) (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), (x * y));
	double tmp;
	if ((a * b) <= -2e+134) {
		tmp = t_1;
	} else if ((a * b) <= 2e+123) {
		tmp = fma(0.0625, (t * z), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), Float64(x * y))
	tmp = 0.0
	if (Float64(a * b) <= -2e+134)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+123)
		tmp = fma(0.0625, Float64(t * z), fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+134], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+123], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999984e134 or 1.99999999999999996e123 < (*.f64 a b)

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 89.1

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

   8. Simplified 89.1%

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

   if -1.99999999999999984e134 < (*.f64 a b) < 1.99999999999999996e123

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.3

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

   5. Simplified 93.3%

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

Alternative 8: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (* x y))))
   (if (<= (* a b) -2e+134)
     t_1
     (if (<= (* a b) 2e+95) (fma (* 0.0625 z) t c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(a, (b * -0.25), (x * y));
	double tmp;
	if ((a * b) <= -2e+134) {
		tmp = t_1;
	} else if ((a * b) <= 2e+95) {
		tmp = fma((0.0625 * z), t, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(a, Float64(b * -0.25), Float64(x * y))
	tmp = 0.0
	if (Float64(a * b) <= -2e+134)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+95)
		tmp = fma(Float64(0.0625 * z), t, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+134], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+95], N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999984e134 or 2.00000000000000004e95 < (*.f64 a b)

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 87.5

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

   8. Simplified 87.5%

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

   if -1.99999999999999984e134 < (*.f64 a b) < 2.00000000000000004e95

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]

      2. *-lowering-*.f64 67.0

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

   5. Simplified 67.0%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + c \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{16} \cdot t\right)} + c \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(z \cdot \frac{1}{16}\right) \cdot t} + c \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \frac{1}{16}, t, c\right)} \]

      5. *-lowering-*.f64 67.0

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

   7. Applied egg-rr 67.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* t z))))
   (if (<= (* t z) -1.5e+115) t_1 (if (<= (* t z) 1e+106) (fma y x c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.0625 * (t * z);
	double tmp;
	if ((t * z) <= -1.5e+115) {
		tmp = t_1;
	} else if ((t * z) <= 1e+106) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1.5e+115)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+106)
		tmp = fma(y, x, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.5e+115], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+106], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.5e115 or 1.00000000000000009e106 < (*.f64 z t)

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]

      2. *-lowering-*.f64 70.8

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

   5. Simplified 70.8%

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

   if -1.5e115 < (*.f64 z t) < 1.00000000000000009e106

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 62.9

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

   5. Simplified 62.9%

      \[\leadsto \color{blue}{x \cdot y} + c \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 62.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]

   7. Applied egg-rr 62.9%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.06 \cdot 10^{+45}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2.25 \cdot 10^{+101}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1.06e+45) (* x y) (if (<= (* x y) 2.25e+101) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1.06e+45) {
		tmp = x * y;
	} else if ((x * y) <= 2.25e+101) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x * y) <= (-1.06d+45)) then
        tmp = x * y
    else if ((x * y) <= 2.25d+101) then
        tmp = c
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1.06e+45) {
		tmp = x * y;
	} else if ((x * y) <= 2.25e+101) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -1.06e+45:
		tmp = x * y
	elif (x * y) <= 2.25e+101:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1.06e+45)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 2.25e+101)
		tmp = c;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -1.06e+45)
		tmp = x * y;
	elseif ((x * y) <= 2.25e+101)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.06e+45], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2.25e+101], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.06e45 or 2.2500000000000001e101 < (*.f64 x y)

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 61.9

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

   5. Simplified 61.9%

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

   if -1.06e45 < (*.f64 x y) < 2.2500000000000001e101

   1. Initial program 99.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c} \]
   4. Step-by-step derivation

   5. Simplified 28.0%

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

Alternative 11: 49.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, c\right) \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 (fma y x c))

C

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

Julia

function code(x, y, z, t, a, b, c)
	return fma(y, x, c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(y * x + c), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot y} + c \]
4. Step-by-step derivation

   1. *-lowering-*.f64 47.4

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

5. Simplified 47.4%

   \[\leadsto \color{blue}{x \cdot y} + c \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 47.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]

7. Applied egg-rr 47.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right)} \]
8. Add Preprocessing

Alternative 12: 22.5% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ c \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 c)

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c
end function

Java

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

Python

def code(x, y, z, t, a, b, c):
	return c

Julia

function code(x, y, z, t, a, b, c)
	return c
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c} \]
4. Step-by-step derivation

5. Simplified 21.8%

   \[\leadsto \color{blue}{c} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))