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

Percentage Accurate: 97.7% → 98.8%

Time: 9.0s

Alternatives: 18

Speedup: 1.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 (* y x))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -5e+154)
     t_1
     (if (<= t_2 -4000000000000.0)
       (fma y x c)
       (if (<= t_2 1e+137)
         (fma -0.25 (* b a) c)
         (if (<= t_2 INFINITY) t_1 (fma -0.25 (* b a) (* y x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, (y * x));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -5e+154) {
		tmp = t_1;
	} else if (t_2 <= -4000000000000.0) {
		tmp = fma(y, x, c);
	} else if (t_2 <= 1e+137) {
		tmp = fma(-0.25, (b * a), c);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(-0.25, (b * a), (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -5e+154)
		tmp = t_1;
	elseif (t_2 <= -4000000000000.0)
		tmp = fma(y, x, c);
	elseif (t_2 <= 1e+137)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+154], t$95$1, If[LessEqual[t$95$2, -4000000000000.0], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$2, 1e+137], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.00000000000000004e154 or 1e137 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < +inf.0

   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 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 85.7%

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

   if -5.00000000000000004e154 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -4e12

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.8%

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

   if -4e12 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1e137

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

   if +inf.0 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 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 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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 57.1%

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

Alternative 3: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_1 -5e+154)
     (fma (* t z) 0.0625 (* y x))
     (if (<= t_1 -4000000000000.0)
       (fma y x c)
       (if (<= t_1 1e+137)
         (fma -0.25 (* b a) c)
         (fma y x (* 0.0625 (* t z))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_1 <= -5e+154) {
		tmp = fma((t * z), 0.0625, (y * x));
	} else if (t_1 <= -4000000000000.0) {
		tmp = fma(y, x, c);
	} else if (t_1 <= 1e+137) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = fma(y, x, (0.0625 * (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_1 <= -5e+154)
		tmp = fma(Float64(t * z), 0.0625, Float64(y * x));
	elseif (t_1 <= -4000000000000.0)
		tmp = fma(y, x, c);
	elseif (t_1 <= 1e+137)
		tmp = fma(-0.25, Float64(b * a), c);
	else
		tmp = fma(y, x, Float64(0.0625 * Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+154], N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4000000000000.0], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$1, 1e+137], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], N[(y * x + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.5%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 86.1%

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

   if -5.00000000000000004e154 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -4e12

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.8%

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

   if -4e12 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1e137

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

   if 1e137 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 89.8%

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

   3. Taylor expanded in a around 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 81.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\\ \mathbf{elif}\;x \cdot y + \frac{z \cdot t}{16} \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y + \frac{z \cdot t}{16} \leq 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* t 0.0625) z c)))
   (if (<= t_1 -2e-63)
     t_2
     (if (<= t_1 0.0)
       (fma y x (* (* b a) -0.25))
       (if (<= t_1 1e+37) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((t * 0.0625), z, c);
	double tmp;
	if (t_1 <= -2e-63) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(y, x, ((b * a) * -0.25));
	} else if (t_1 <= 1e+37) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(t * 0.0625), z, c)
	tmp = 0.0
	if (t_1 <= -2e-63)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(y, x, Float64(Float64(b * a) * -0.25));
	elseif (t_1 <= 1e+37)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-63], t$95$2, If[LessEqual[t$95$1, 0.0], N[(y * x + N[(N[(b * a), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+37], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.00000000000000013e-63 or 9.99999999999999954e36 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.1%

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

   10. Applied rewrites 77.1%

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

   if -2.00000000000000013e-63 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -0.0

   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. 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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 73.7%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{b \cdot a}, y \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.7%

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

   if -0.0 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999954e36

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.2%

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

Alternative 5: 64.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* t 0.0625) z c)))
   (if (<= t_1 -2e-63)
     t_2
     (if (<= t_1 0.0)
       (fma -0.25 (* b a) (* y x))
       (if (<= t_1 1e+37) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((t * 0.0625), z, c);
	double tmp;
	if (t_1 <= -2e-63) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(-0.25, (b * a), (y * x));
	} else if (t_1 <= 1e+37) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(t * 0.0625), z, c)
	tmp = 0.0
	if (t_1 <= -2e-63)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	elseif (t_1 <= 1e+37)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-63], t$95$2, If[LessEqual[t$95$1, 0.0], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+37], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.00000000000000013e-63 or 9.99999999999999954e36 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.7%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.1%

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

   10. Applied rewrites 77.1%

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

   if -2.00000000000000013e-63 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -0.0

   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. 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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 73.7%

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

   if -0.0 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999954e36

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.2%

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

Alternative 6: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \left(t \cdot z\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -2e-40)
     (fma (* 0.0625 z) t (fma y x c))
     (if (<= t_1 1e+62)
       (fma y x (fma -0.25 (* b a) c))
       (fma -0.25 (* b a) (* (* t z) 0.0625))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -2e-40) {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	} else if (t_1 <= 1e+62) {
		tmp = fma(y, x, fma(-0.25, (b * a), c));
	} else {
		tmp = fma(-0.25, (b * a), ((t * z) * 0.0625));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -2e-40)
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	elseif (t_1 <= 1e+62)
		tmp = fma(y, x, fma(-0.25, Float64(b * a), c));
	else
		tmp = fma(-0.25, Float64(b * a), Float64(Float64(t * z) * 0.0625));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-40], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+62], N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.9999999999999999e-40

   1. Initial program 92.1%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 86.5%

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

   if -1.9999999999999999e-40 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.00000000000000004e62

   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 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if 1.00000000000000004e62 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 94.7%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 84.9%

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

Alternative 7: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-40} \lor \neg \left(t\_1 \leq 5000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e-40) (not (<= t_1 5000000000000.0)))
     (fma y x (fma (* t z) 0.0625 c))
     (fma y x (fma -0.25 (* b a) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e-40) || !(t_1 <= 5000000000000.0)) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma(y, x, fma(-0.25, (b * a), c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e-40) || !(t_1 <= 5000000000000.0))
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(y, x, fma(-0.25, Float64(b * a), c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-40], N[Not[LessEqual[t$95$1, 5000000000000.0]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.9999999999999999e-40 or 5e12 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.5%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.9999999999999999e-40 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5e12

   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 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{-40} \lor \neg \left(\frac{z \cdot t}{16} \leq 5000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+98} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e+98) (not (<= t_1 2e+178)))
     (fma (* t 0.0625) z c)
     (fma y x (fma -0.25 (* b a) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e+98) || !(t_1 <= 2e+178)) {
		tmp = fma((t * 0.0625), z, c);
	} else {
		tmp = fma(y, x, fma(-0.25, (b * a), c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e+98) || !(t_1 <= 2e+178))
		tmp = fma(Float64(t * 0.0625), z, c);
	else
		tmp = fma(y, x, fma(-0.25, Float64(b * a), c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+98], N[Not[LessEqual[t$95$1, 2e+178]], $MachinePrecision]], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e98 or 2.0000000000000001e178 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 90.1%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.0%

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

   10. Applied rewrites 86.0%

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

   if -2e98 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000001e178

   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 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{+98} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+98} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e+98) (not (<= t_1 2e+178)))
     (fma (* t 0.0625) z c)
     (fma -0.25 (* b a) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e+98) || !(t_1 <= 2e+178)) {
		tmp = fma((t * 0.0625), z, c);
	} else {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e+98) || !(t_1 <= 2e+178))
		tmp = fma(Float64(t * 0.0625), z, c);
	else
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+98], N[Not[LessEqual[t$95$1, 2e+178]], $MachinePrecision]], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e98 or 2.0000000000000001e178 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 90.1%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.0%

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

   10. Applied rewrites 86.0%

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

   if -2e98 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000001e178

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{+98} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(-0.25, b \cdot a, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -2e-40)
     (fma (* 0.0625 z) t (fma y x c))
     (if (<= t_1 5000000000000.0)
       (fma y x (fma -0.25 (* b a) c))
       (fma y x (fma (* t z) 0.0625 c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -2e-40) {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	} else if (t_1 <= 5000000000000.0) {
		tmp = fma(y, x, fma(-0.25, (b * a), c));
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -2e-40)
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	elseif (t_1 <= 5000000000000.0)
		tmp = fma(y, x, fma(-0.25, Float64(b * a), c));
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-40], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5000000000000.0], N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.9999999999999999e-40

   1. Initial program 92.1%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 86.5%

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

   if -1.9999999999999999e-40 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5e12

   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 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

   if 5e12 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   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 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 11: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-40} \lor \neg \left(t\_1 \leq 5000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e-40) (not (<= t_1 5000000000000.0)))
     (fma (* t 0.0625) z c)
     (fma -0.25 (* b a) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e-40) || !(t_1 <= 5000000000000.0)) {
		tmp = fma((t * 0.0625), z, c);
	} else {
		tmp = fma(-0.25, (b * a), c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e-40) || !(t_1 <= 5000000000000.0))
		tmp = fma(Float64(t * 0.0625), z, c);
	else
		tmp = fma(-0.25, Float64(b * a), c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-40], N[Not[LessEqual[t$95$1, 5000000000000.0]], $MachinePrecision]], N[(N[(t * 0.0625), $MachinePrecision] * z + c), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.9999999999999999e-40 or 5e12 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.5%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.9%

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

   10. Applied rewrites 75.9%

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

   if -1.9999999999999999e-40 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5e12

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{-40} \lor \neg \left(\frac{z \cdot t}{16} \leq 5000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+205} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e+205) (not (<= t_1 2e+178)))
     (* (* t z) 0.0625)
     (fma -0.25 (* b a) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e+205) || !(t_1 <= 2e+178)) {
		tmp = (t * z) * 0.0625;
	} else {
		tmp = fma(-0.25, (b * a), c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e+205) || !(t_1 <= 2e+178))
		tmp = Float64(Float64(t * z) * 0.0625);
	else
		tmp = fma(-0.25, Float64(b * a), c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+205], N[Not[LessEqual[t$95$1, 2e+178]], $MachinePrecision]], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.0000000000000002e205 or 2.0000000000000001e178 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 84.1

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

   8. Applied rewrites 84.1%

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

   if -5.0000000000000002e205 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000001e178

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{+205} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+140} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -1e+140) (not (<= t_1 2e+178)))
     (* (* t z) 0.0625)
     (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -1e+140) || !(t_1 <= 2e+178)) {
		tmp = (t * z) * 0.0625;
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -1e+140) || !(t_1 <= 2e+178))
		tmp = Float64(Float64(t * z) * 0.0625);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+140], N[Not[LessEqual[t$95$1, 2e+178]], $MachinePrecision]], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.00000000000000006e140 or 2.0000000000000001e178 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if -1.00000000000000006e140 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000001e178

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.5%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -1 \cdot 10^{+140} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+190} \lor \neg \left(t\_1 \leq 10^{+111}\right):\\ \;\;\;\;\left(-0.25 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+190) (not (<= t_1 1e+111)))
     (* (* -0.25 a) b)
     (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e+190) || !(t_1 <= 1e+111)) {
		tmp = (-0.25 * a) * b;
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e+190) || !(t_1 <= 1e+111))
		tmp = Float64(Float64(-0.25 * a) * b);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+190], N[Not[LessEqual[t$95$1, 1e+111]], $MachinePrecision]], N[(N[(-0.25 * a), $MachinePrecision] * b), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.00000000000000036e190 or 9.99999999999999957e110 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -5.00000000000000036e190 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999957e110

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 55.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+190} \lor \neg \left(\frac{a \cdot b}{4} \leq 10^{+111}\right):\\ \;\;\;\;\left(-0.25 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625, t \cdot z, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+94)
   (fma y x (fma (* t z) 0.0625 c))
   (if (<= (* x y) 1e+101)
     (fma (* -0.25 b) a (fma 0.0625 (* t z) c))
     (fma (* 0.0625 z) t (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) <= -2e+94) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else if ((x * y) <= 1e+101) {
		tmp = fma((-0.25 * b), a, fma(0.0625, (t * z), c));
	} else {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+94)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	elseif (Float64(x * y) <= 1e+101)
		tmp = fma(Float64(-0.25 * b), a, fma(0.0625, Float64(t * z), c));
	else
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+94], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+101], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(0.0625 * N[(t * z), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e94

   1. Initial program 88.2%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -2e94 < (*.f64 x y) < 9.9999999999999998e100

   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. fp-cancel-sub-sign-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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.9%

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

   if 9.9999999999999998e100 < (*.f64 x y)

   1. Initial program 89.4%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 88.9%

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

Alternative 16: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c)))
   (if (<= (* x y) -2e+94)
     (fma y x t_1)
     (if (<= (* x y) 1e+101)
       (fma -0.25 (* b a) t_1)
       (fma (* 0.0625 z) t (fma y x c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, c);
	double tmp;
	if ((x * y) <= -2e+94) {
		tmp = fma(y, x, t_1);
	} else if ((x * y) <= 1e+101) {
		tmp = fma(-0.25, (b * a), t_1);
	} else {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+94)
		tmp = fma(y, x, t_1);
	elseif (Float64(x * y) <= 1e+101)
		tmp = fma(-0.25, Float64(b * a), t_1);
	else
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+94], N[(y * x + t$95$1), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+101], N[(-0.25 * N[(b * a), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e94

   1. Initial program 88.2%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -2e94 < (*.f64 x y) < 9.9999999999999998e100

   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. fp-cancel-sub-sign-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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 9.9999999999999998e100 < (*.f64 x y)

   1. Initial program 89.4%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 88.9%

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

Alternative 17: 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 96.5%

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

3. Taylor expanded in z around 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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 67.5

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 48.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 43.9%

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

Alternative 18: 28.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 21.7

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

8. Applied rewrites 21.7%

   \[\leadsto \color{blue}{y \cdot x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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))