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

Percentage Accurate: 97.8% → 98.5%

Time: 12.4s

Alternatives: 13

Speedup: 0.6×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \mathsf{fma}\left(a, b \cdot 0.25, 0 - c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) INFINITY)
   (fma (* t 0.0625) z (- (* x y) (fma a (* b 0.25) (- 0.0 c))))
   (* 0.0625 (* z t))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) <= ((double) INFINITY)) {
		tmp = fma((t * 0.0625), z, ((x * y) - fma(a, (b * 0.25), (0.0 - c))));
	} else {
		tmp = 0.0625 * (z * t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) <= Inf)
		tmp = fma(Float64(t * 0.0625), z, Float64(Float64(x * y) - fma(a, Float64(b * 0.25), Float64(0.0 - c))));
	else
		tmp = Float64(0.0625 * Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(N[(x * y), $MachinePrecision] - N[(a * N[(b * 0.25), $MachinePrecision] + N[(0.0 - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) < +inf.0

   1. Initial program 100.0%

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

      1. associate-+l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{t}{16}\right), \color{blue}{z}, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{1}{16}\right)\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\left(x \cdot y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(a \cdot \frac{b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(\frac{b}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(b \cdot \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \left(\frac{1}{4}\right)\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f64 100.0%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \mathsf{neg.f64}\left(c\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, 0\right) \]

      3. *-lowering-*.f64 57.6%

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), 0\right) \]

   5. Simplified 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(t \cdot z\right), \color{blue}{\frac{1}{16}}\right) \]

      4. *-lowering-*.f64 57.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, z\right), \frac{1}{16}\right) \]

   7. Applied egg-rr 57.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \mathsf{fma}\left(a, b \cdot 0.25, 0 - c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -100:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -100.0)
   (fma y x c)
   (if (<= (* x y) 1e-230)
     (fma (* t 0.0625) z c)
     (if (<= (* x y) 1e+151) (fma a (* b -0.25) c) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -100.0) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 1e-230) {
		tmp = fma((t * 0.0625), z, c);
	} else if ((x * y) <= 1e+151) {
		tmp = fma(a, (b * -0.25), c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -100.0)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 1e-230)
		tmp = fma(Float64(t * 0.0625), z, c);
	elseif (Float64(x * y) <= 1e+151)
		tmp = fma(a, Float64(b * -0.25), c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -100.0], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-230], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+151], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -100 or 1.00000000000000002e151 < (*.f64 x y)

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y + 0\right), c\right) \]

      2. accelerator-lowering-fma.f64 71.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(x, y, 0\right), c\right) \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. accelerator-lowering-fma.f64 71.7%

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, c\right) \]

   7. Applied egg-rr 71.7%

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

   if -100 < (*.f64 x y) < 1.00000000000000005e-230

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + 0\right), c\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \left(t \cdot z\right), 0\right), c\right) \]

      3. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), 0\right), c\right) \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), \color{blue}{z}, c\right) \]

      5. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, c\right) \]

   7. Applied egg-rr 73.4%

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

   if 1.00000000000000005e-230 < (*.f64 x y) < 1.00000000000000002e151

   1. Initial program 98.3%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 80.7%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 80.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{c}\right) \]
   7. Step-by-step derivation

   8. Simplified 65.5%

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

Alternative 3: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+146)
   (fma 0.0625 (* z t) (fma x y c))
   (if (<= (* x y) 1e-22)
     (fma 0.0625 (* z t) (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) <= -2e+146) {
		tmp = fma(0.0625, (z * t), fma(x, y, c));
	} else if ((x * y) <= 1e-22) {
		tmp = fma(0.0625, (z * t), 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) <= -2e+146)
		tmp = fma(0.0625, Float64(z * t), fma(x, y, c));
	elseif (Float64(x * y) <= 1e-22)
		tmp = fma(0.0625, Float64(z * t), 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], -2e+146], N[(0.0625 * N[(z * t), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-22], N[(0.0625 * N[(z * t), $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) < -1.99999999999999987e146

   1. Initial program 90.0%

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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), \left(c + x \cdot y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(x \cdot y + c\right)\right) \]

      7. accelerator-lowering-fma.f64 93.3%

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 93.3%

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

   if -1.99999999999999987e146 < (*.f64 x y) < 1e-22

   1. Initial program 98.6%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, \left(c + \frac{-1}{4} \cdot \left(a \cdot b\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), \left(c + \frac{-1}{4} \cdot \left(a \cdot b\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + c\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(\left(a \cdot b\right) \cdot \frac{-1}{4} + c\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(a \cdot \left(b \cdot \frac{-1}{4}\right) + c\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(a \cdot \left(\frac{-1}{4} \cdot b\right) + c\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(a, \left(\frac{-1}{4} \cdot b\right), c\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(a, \left(b \cdot \frac{-1}{4}\right), c\right)\right) \]

      13. *-lowering-*.f64 90.5%

          \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), c\right)\right) \]

   5. Simplified 90.5%

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

   if 1e-22 < (*.f64 x y)

   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 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 89.8%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 89.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(a, b \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -0.0002)
   (fma (* t 0.0625) z (* x y))
   (if (<= (* x y) 1e-65) (fma (* t 0.0625) z 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 ((x * y) <= -0.0002) {
		tmp = fma((t * 0.0625), z, (x * y));
	} else if ((x * y) <= 1e-65) {
		tmp = fma((t * 0.0625), z, 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(x * y) <= -0.0002)
		tmp = fma(Float64(t * 0.0625), z, Float64(x * y));
	elseif (Float64(x * y) <= 1e-65)
		tmp = fma(Float64(t * 0.0625), z, 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[(x * y), $MachinePrecision], -0.0002], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-65], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000001e-4

   1. Initial program 95.2%

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

      1. associate-+l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{t}{16}\right), \color{blue}{z}, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{1}{16}\right)\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\left(x \cdot y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(a \cdot \frac{b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(\frac{b}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(b \cdot \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \left(\frac{1}{4}\right)\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f64 95.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \mathsf{neg.f64}\left(c\right)\right)\right)\right) \]

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \color{blue}{\left(x \cdot y\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{*.f64}\left(x, y\right)\right) \]

   7. Simplified 70.3%

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

   if -2.0000000000000001e-4 < (*.f64 x y) < 9.99999999999999923e-66

   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 z around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + 0\right), c\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \left(t \cdot z\right), 0\right), c\right) \]

      3. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), 0\right), c\right) \]

   5. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), \color{blue}{z}, c\right) \]

      5. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, c\right) \]

   7. Applied egg-rr 71.1%

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

   if 9.99999999999999923e-66 < (*.f64 x y)

   1. Initial program 97.6%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 89.2%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 89.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 77.8%

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   8. Simplified 77.8%

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

Alternative 5: 66.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) (* x y))))
   (if (<= (* x y) -100.0)
     t_1
     (if (<= (* x y) 1e-65) (fma (* t 0.0625) z 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 ((x * y) <= -100.0) {
		tmp = t_1;
	} else if ((x * y) <= 1e-65) {
		tmp = fma((t * 0.0625), z, 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(x * y) <= -100.0)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-65)
		tmp = fma(Float64(t * 0.0625), z, 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[(x * y), $MachinePrecision], -100.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-65], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -100 or 9.99999999999999923e-66 < (*.f64 x y)

   1. Initial program 96.6%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 84.4%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 84.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 73.7%

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   8. Simplified 73.7%

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

   if -100 < (*.f64 x y) < 9.99999999999999923e-66

   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 z around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + 0\right), c\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \left(t \cdot z\right), 0\right), c\right) \]

      3. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), 0\right), c\right) \]

   5. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), \color{blue}{z}, c\right) \]

      5. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, c\right) \]

   7. Applied egg-rr 71.1%

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

Alternative 6: 83.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* z t) (fma x y c))))
   (if (<= t -1.26e-7)
     t_1
     (if (<= t 1.45e+123) (fma a (* b -0.25) (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(0.0625, (z * t), fma(x, y, c));
	double tmp;
	if (t <= -1.26e-7) {
		tmp = t_1;
	} else if (t <= 1.45e+123) {
		tmp = fma(a, (b * -0.25), fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(z * t), fma(x, y, c))
	tmp = 0.0
	if (t <= -1.26e-7)
		tmp = t_1;
	elseif (t <= 1.45e+123)
		tmp = fma(a, Float64(b * -0.25), 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[(0.0625 * N[(z * t), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.26e-7], t$95$1, If[LessEqual[t, 1.45e+123], N[(a * N[(b * -0.25), $MachinePrecision] + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2599999999999999e-7 or 1.45000000000000005e123 < t

   1. Initial program 95.0%

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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), \left(c + x \cdot y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(x \cdot y + c\right)\right) \]

      7. accelerator-lowering-fma.f64 81.5%

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 81.5%

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

   if -1.2599999999999999e-7 < t < 1.45000000000000005e123

   1. Initial program 98.7%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 88.5%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 88.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= a -2.6e+121)
   (fma a (* b -0.25) c)
   (if (<= a 1.05e+31)
     (fma 0.0625 (* z t) (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 <= -2.6e+121) {
		tmp = fma(a, (b * -0.25), c);
	} else if (a <= 1.05e+31) {
		tmp = fma(0.0625, (z * t), 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 (a <= -2.6e+121)
		tmp = fma(a, Float64(b * -0.25), c);
	elseif (a <= 1.05e+31)
		tmp = fma(0.0625, Float64(z * t), 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[a, -2.6e+121], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[a, 1.05e+31], N[(0.0625 * N[(z * t), $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 a < -2.5999999999999999e121

   1. Initial program 93.1%

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 80.1%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 80.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{c}\right) \]
   7. Step-by-step derivation

   8. Simplified 73.2%

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

   if -2.5999999999999999e121 < a < 1.04999999999999989e31

   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 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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), \left(c + x \cdot y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \left(x \cdot y + c\right)\right) \]

      7. accelerator-lowering-fma.f64 86.7%

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, z\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 86.7%

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

   if 1.04999999999999989e31 < a

   1. Initial program 98.4%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 78.5%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{\left(x \cdot y\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   8. Simplified 67.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z \cdot t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma a (* b -0.25) c)))
   (if (<= a -6.5e+121) t_1 (if (<= a 1.72e+43) (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(a, (b * -0.25), c);
	double tmp;
	if (a <= -6.5e+121) {
		tmp = t_1;
	} else if (a <= 1.72e+43) {
		tmp = fma(y, x, 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), c)
	tmp = 0.0
	if (a <= -6.5e+121)
		tmp = t_1;
	elseif (a <= 1.72e+43)
		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[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[a, -6.5e+121], t$95$1, If[LessEqual[a, 1.72e+43], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.50000000000000019e121 or 1.7199999999999999e43 < a

   1. Initial program 97.8%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}, \left(c + x \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right), \left(c + x \cdot y\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \left(x \cdot y + c\right)\right) \]

      11. accelerator-lowering-fma.f64 79.4%

          \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \mathsf{fma.f64}\left(x, y, c\right)\right) \]

   5. Simplified 79.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{-1}{4}\right), \color{blue}{c}\right) \]
   7. Step-by-step derivation

   8. Simplified 57.7%

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

   if -6.50000000000000019e121 < a < 1.7199999999999999e43

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y + 0\right), c\right) \]

      2. accelerator-lowering-fma.f64 60.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(x, y, 0\right), c\right) \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. accelerator-lowering-fma.f64 60.2%

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, c\right) \]

   7. Applied egg-rr 60.2%

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

Alternative 9: 53.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t \leq -57000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.0625 (* z t))))
   (if (<= t -57000000000.0) t_1 (if (<= t 1.05e+150) (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 * (z * t);
	double tmp;
	if (t <= -57000000000.0) {
		tmp = t_1;
	} else if (t <= 1.05e+150) {
		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(z * t))
	tmp = 0.0
	if (t <= -57000000000.0)
		tmp = t_1;
	elseif (t <= 1.05e+150)
		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[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -57000000000.0], t$95$1, If[LessEqual[t, 1.05e+150], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.7e10 or 1.04999999999999999e150 < t

   1. Initial program 94.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. +-rgt-identity N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \color{blue}{\left(t \cdot z\right)}, 0\right) \]

      3. *-lowering-*.f64 51.6%

         \[\leadsto \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(t, \color{blue}{z}\right), 0\right) \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, t \cdot z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(t \cdot z\right), \color{blue}{\frac{1}{16}}\right) \]

      4. *-lowering-*.f64 51.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, z\right), \frac{1}{16}\right) \]

   7. Applied egg-rr 51.6%

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

   if -5.7e10 < t < 1.04999999999999999e150

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y + 0\right), c\right) \]

      2. accelerator-lowering-fma.f64 61.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(x, y, 0\right), c\right) \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. accelerator-lowering-fma.f64 61.4%

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, c\right) \]

   7. Applied egg-rr 61.4%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -57000000000:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ \mathbf{if}\;a \leq -2.15 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* a (* b -0.25))))
   (if (<= a -2.15e+175) t_1 (if (<= a 4.8e+77) (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 = a * (b * -0.25);
	double tmp;
	if (a <= -2.15e+175) {
		tmp = t_1;
	} else if (a <= 4.8e+77) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(a * Float64(b * -0.25))
	tmp = 0.0
	if (a <= -2.15e+175)
		tmp = t_1;
	elseif (a <= 4.8e+77)
		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[(a * N[(b * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.15e+175], t$95$1, If[LessEqual[a, 4.8e+77], N[(y * x + c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.14999999999999992e175 or 4.7999999999999997e77 < a

   1. Initial program 98.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(b \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      6. *-lowering-*.f64 49.6%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right)\right) \]

   5. Simplified 49.6%

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

   if -2.14999999999999992e175 < a < 4.7999999999999997e77

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, c\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y + 0\right), c\right) \]

      2. accelerator-lowering-fma.f64 59.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(x, y, 0\right), c\right) \]

   5. Simplified 59.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. accelerator-lowering-fma.f64 59.1%

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, c\right) \]

   7. Applied egg-rr 59.1%

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

Alternative 11: 36.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+135}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 0.00106:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -7.6e+135) c (if (<= c 0.00106) (* x y) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -7.6e+135) {
		tmp = c;
	} else if (c <= 0.00106) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (c <= (-7.6d+135)) then
        tmp = c
    else if (c <= 0.00106d0) then
        tmp = x * y
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -7.6e+135) {
		tmp = c;
	} else if (c <= 0.00106) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if c <= -7.6e+135:
		tmp = c
	elif c <= 0.00106:
		tmp = x * y
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= -7.6e+135)
		tmp = c;
	elseif (c <= 0.00106)
		tmp = Float64(x * y);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (c <= -7.6e+135)
		tmp = c;
	elseif (c <= 0.00106)
		tmp = x * y;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, -7.6e+135], c, If[LessEqual[c, 0.00106], N[(x * y), $MachinePrecision], c]]

Derivation

1. Split input into 2 regimes

2. if c < -7.6000000000000003e135 or 0.00105999999999999996 < c

   1. Initial program 97.8%

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

   3. Taylor expanded in c around inf

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

   5. Simplified 50.9%

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

   if -7.6000000000000003e135 < c < 0.00105999999999999996

   1. Initial program 96.9%

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

      1. associate-+l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{t}{16}\right), \color{blue}{z}, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(t \cdot \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{1}{16}\right)\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\left(x \cdot y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} - c\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{a \cdot b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(a \cdot \frac{b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(\frac{b}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \left(b \cdot \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \left(\frac{1}{4}\right)\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f64 96.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{16}\right), z, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{fma.f64}\left(a, \mathsf{*.f64}\left(b, \frac{1}{4}\right), \mathsf{neg.f64}\left(c\right)\right)\right)\right) \]

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 41.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 41.3%

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

Alternative 12: 48.3% 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 97.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, c\right) \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y + 0\right), c\right) \]

   2. accelerator-lowering-fma.f64 52.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(x, y, 0\right), c\right) \]

5. Simplified 52.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} + c \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot y + c \]

   2. *-commutative N/A

      \[\leadsto y \cdot x + c \]

   3. accelerator-lowering-fma.f64 52.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, c\right) \]

7. Applied egg-rr 52.0%

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

Alternative 13: 22.2% 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 97.3%

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

3. Taylor expanded in c around inf

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

5. Simplified 21.4%

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

Reproduce

herbie shell --seed 2024193 
(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))