tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 25.4s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tan-sum N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-tan.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-tan.f64 99.8

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

4. Applied egg-rr 99.8%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.02)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 1e-14)
     (+
      x
      (-
       (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))
       (fma a (* a (* a 0.3333333333333333)) a)))
     (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 1e-14) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - fma(a, (a * (a * 0.3333333333333333)), a));
	} else {
		tmp = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 1e-14)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(a, Float64(a * Float64(a * 0.3333333333333333)), a)));
	else
		tmp = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 80.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   if -0.0200000000000000004 < (tan.f64 a) < 9.99999999999999999e-15

   1. Initial program 80.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 80.0

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 80.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]
   6. Step-by-step derivation

      1. tan-sum N/A

         \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y} + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      3. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \color{blue}{\tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      5. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y} \cdot \tan z} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      6. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \color{blue}{\tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y \cdot \tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      8. lift--.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot \frac{1}{3}\right), a\right)\right) \]

      9. lift-/.f64 99.9

         \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right) \]

   if 9.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 82.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x + \left(\tan \color{blue}{\left(y + z\right)} - \tan a\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      3. lift-tan.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\tan a}\right) \]

      4. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) + \left(\mathsf{neg}\left(\tan a\right)\right)\right)} \]

      5. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

      6. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto x + \left(\color{blue}{\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right)} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{\frac{1}{\cos \left(y + z\right)}}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\frac{1}{\color{blue}{\cos \left(y + z\right)}}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right) \]

      12. lower-sin.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \color{blue}{\sin \left(y + z\right)}, \mathsf{neg}\left(\tan a\right)\right) \]

      13. lower-neg.f64 82.3

          \[\leadsto x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), \color{blue}{-\tan a}\right) \]

   4. Applied egg-rr 82.3%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto x + \left(\tan \color{blue}{\left(y + z\right)} - \tan a\right) \]

   2. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   3. lift-tan.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\tan a}\right) \]

   4. sub-neg N/A

      \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) + \left(\mathsf{neg}\left(\tan a\right)\right)\right)} \]

   5. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

   6. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

   8. associate-/r/ N/A

      \[\leadsto x + \left(\color{blue}{\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right)} + \left(\mathsf{neg}\left(\tan a\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(\color{blue}{\frac{1}{\cos \left(y + z\right)}}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right) \]

   11. lower-cos.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(\frac{1}{\color{blue}{\cos \left(y + z\right)}}, \sin \left(y + z\right), \mathsf{neg}\left(\tan a\right)\right) \]

   12. lower-sin.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \color{blue}{\sin \left(y + z\right)}, \mathsf{neg}\left(\tan a\right)\right) \]

   13. lower-neg.f64 80.7

       \[\leadsto x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), \color{blue}{-\tan a}\right) \]

4. Applied egg-rr 80.7%

   \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 50.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9e-8)
   (+ x (sin y))
   (if (<= a 1.55)
     (+
      x
      (-
       (tan (+ y z))
       (fma
        (fma
         (* a a)
         (fma (* a a) 0.05396825396825397 0.13333333333333333)
         0.3333333333333333)
        (* a (* a a))
        a)))
     (+ x (/ (sin y) (fma y (* y -0.5) 1.0))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9e-8) {
		tmp = x + sin(y);
	} else if (a <= 1.55) {
		tmp = x + (tan((y + z)) - fma(fma((a * a), fma((a * a), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = x + (sin(y) / fma(y, (y * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9e-8)
		tmp = Float64(x + sin(y));
	elseif (a <= 1.55)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(Float64(a * a), fma(Float64(a * a), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(sin(y) / fma(y, Float64(y * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9e-8], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] / N[(y * N[(y * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.90000000000000014e-8

   1. Initial program 78.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 3.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 3.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 3.7

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 22.0

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]
   13. Step-by-step derivation

   14. Simplified 22.9%

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]

   if -1.90000000000000014e-8 < a < 1.55000000000000004

   1. Initial program 80.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) + 1\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a + 1 \cdot a\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{a \cdot \left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)} + 1 \cdot a\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot {a}^{2}\right) \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)} + 1 \cdot a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left(a \cdot {a}^{2}\right)} + 1 \cdot a\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left(a \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right), a \cdot {a}^{2}, a\right)}\right) \]

   5. Simplified 79.9%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]

   if 1.55000000000000004 < a

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 1.1

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 1.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 1.1

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 20.9

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1 + \frac{-1}{2} \cdot {y}^{2}}} + x \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\frac{-1}{2} \cdot {y}^{2} + 1}} + x \]

       2. *-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{{y}^{2} \cdot \frac{-1}{2}} + 1} + x \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{2} + 1} + x \]

       4. associate-*l* N/A

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot \left(y \cdot \frac{-1}{2}\right)} + 1} + x \]

       5. *-commutative N/A

          \[\leadsto \frac{\sin y}{y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y\right)} + 1} + x \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot y, 1\right)}} + x \]

       7. *-commutative N/A

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{2}}, 1\right)} + x \]

       8. lower-*.f64 20.3

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot -0.5}, 1\right)} + x \]

   14. Simplified 20.3%

       \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}} + x \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9e-8)
   (+ x (sin y))
   (if (<= a 1.55)
     (+
      x
      (-
       (tan (+ y z))
       (fma
        (fma (* a a) 0.13333333333333333 0.3333333333333333)
        (* a (* a a))
        a)))
     (+ x (/ (sin y) (fma y (* y -0.5) 1.0))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9e-8) {
		tmp = x + sin(y);
	} else if (a <= 1.55) {
		tmp = x + (tan((y + z)) - fma(fma((a * a), 0.13333333333333333, 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = x + (sin(y) / fma(y, (y * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9e-8)
		tmp = Float64(x + sin(y));
	elseif (a <= 1.55)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(Float64(a * a), 0.13333333333333333, 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(sin(y) / fma(y, Float64(y * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9e-8], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] / N[(y * N[(y * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.90000000000000014e-8

   1. Initial program 78.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 3.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 3.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 3.7

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 22.0

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]
   13. Step-by-step derivation

   14. Simplified 22.9%

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]

   if -1.90000000000000014e-8 < a < 1.55000000000000004

   1. Initial program 80.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot {a}^{2}\right) \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}, a \cdot {a}^{2}, a\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {a}^{2} + \frac{1}{3}}, a \cdot {a}^{2}, a\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \frac{2}{15}} + \frac{1}{3}, a \cdot {a}^{2}, a\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({a}^{2}, \frac{2}{15}, \frac{1}{3}\right)}, a \cdot {a}^{2}, a\right)\right) \]

      10. unpow2 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{2}{15}, \frac{1}{3}\right), a \cdot {a}^{2}, a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{2}{15}, \frac{1}{3}\right), a \cdot {a}^{2}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \frac{2}{15}, \frac{1}{3}\right), \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \frac{2}{15}, \frac{1}{3}\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 79.9

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   5. Simplified 79.9%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]

   if 1.55000000000000004 < a

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 1.1

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 1.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 1.1

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 20.9

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1 + \frac{-1}{2} \cdot {y}^{2}}} + x \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\frac{-1}{2} \cdot {y}^{2} + 1}} + x \]

       2. *-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{{y}^{2} \cdot \frac{-1}{2}} + 1} + x \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{2} + 1} + x \]

       4. associate-*l* N/A

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot \left(y \cdot \frac{-1}{2}\right)} + 1} + x \]

       5. *-commutative N/A

          \[\leadsto \frac{\sin y}{y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y\right)} + 1} + x \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot y, 1\right)}} + x \]

       7. *-commutative N/A

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{2}}, 1\right)} + x \]

       8. lower-*.f64 20.3

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot -0.5}, 1\right)} + x \]

   14. Simplified 20.3%

       \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}} + x \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9e-8)
   (+ x (sin y))
   (if (<= a 1.55)
     (+ x (- (tan (+ y z)) (fma a (* a (* a 0.3333333333333333)) a)))
     (+ x (/ (sin y) (fma y (* y -0.5) 1.0))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9e-8) {
		tmp = x + sin(y);
	} else if (a <= 1.55) {
		tmp = x + (tan((y + z)) - fma(a, (a * (a * 0.3333333333333333)), a));
	} else {
		tmp = x + (sin(y) / fma(y, (y * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9e-8)
		tmp = Float64(x + sin(y));
	elseif (a <= 1.55)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(a, Float64(a * Float64(a * 0.3333333333333333)), a)));
	else
		tmp = Float64(x + Float64(sin(y) / fma(y, Float64(y * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9e-8], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(a * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] / N[(y * N[(y * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.90000000000000014e-8

   1. Initial program 78.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 3.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 3.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 3.7

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 22.0

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]
   13. Step-by-step derivation

   14. Simplified 22.9%

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]

   if -1.90000000000000014e-8 < a < 1.55000000000000004

   1. Initial program 80.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 79.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 79.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   if 1.55000000000000004 < a

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 1.1

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 1.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 1.1

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 20.9

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1 + \frac{-1}{2} \cdot {y}^{2}}} + x \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\frac{-1}{2} \cdot {y}^{2} + 1}} + x \]

       2. *-commutative N/A

          \[\leadsto \frac{\sin y}{\color{blue}{{y}^{2} \cdot \frac{-1}{2}} + 1} + x \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{2} + 1} + x \]

       4. associate-*l* N/A

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot \left(y \cdot \frac{-1}{2}\right)} + 1} + x \]

       5. *-commutative N/A

          \[\leadsto \frac{\sin y}{y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y\right)} + 1} + x \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot y, 1\right)}} + x \]

       7. *-commutative N/A

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{2}}, 1\right)} + x \]

       8. lower-*.f64 20.3

          \[\leadsto \frac{\sin y}{\mathsf{fma}\left(y, \color{blue}{y \cdot -0.5}, 1\right)} + x \]

   14. Simplified 20.3%

       \[\leadsto \frac{\sin y}{\color{blue}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}} + x \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin y}{\mathsf{fma}\left(y, y \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9e-8)
   (+ x (sin y))
   (if (<= a 1.55)
     (+ x (- (tan (+ y z)) (fma a (* a (* a 0.3333333333333333)) a)))
     (+ x (tan y)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9e-8) {
		tmp = x + sin(y);
	} else if (a <= 1.55) {
		tmp = x + (tan((y + z)) - fma(a, (a * (a * 0.3333333333333333)), a));
	} else {
		tmp = x + tan(y);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9e-8)
		tmp = Float64(x + sin(y));
	elseif (a <= 1.55)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(a, Float64(a * Float64(a * 0.3333333333333333)), a)));
	else
		tmp = Float64(x + tan(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9e-8], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(a * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.90000000000000014e-8

   1. Initial program 78.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 3.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 3.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 3.7

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 3.7%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 22.0

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 22.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]
   13. Step-by-step derivation

   14. Simplified 22.9%

       \[\leadsto \frac{\sin y}{\color{blue}{1}} + x \]

   if -1.90000000000000014e-8 < a < 1.55000000000000004

   1. Initial program 80.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 79.7

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 79.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   if 1.55000000000000004 < a

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 1.1

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 1.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 1.1

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 20.9

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 20.9%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]
   12. Step-by-step derivation

       1. tan-quot N/A

          \[\leadsto \color{blue}{\tan y} + x \]

       2. lift-tan.f64 20.9

          \[\leadsto \color{blue}{\tan y} + x \]

   13. Applied egg-rr 20.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \tan y \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (tan y)))

C

double code(double x, double y, double z, double a) {
	return x + tan(y);
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + tan(y)
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + Math.tan(y);
}

Python

def code(x, y, z, a):
	return x + math.tan(y)

Julia

function code(x, y, z, a)
	return Float64(x + tan(y))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + tan(y);
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

   6. unpow2 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

   9. lower-*.f64 41.0

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

5. Simplified 41.0%

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

   10. cube-mult N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   14. lower-*.f64 31.3

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

8. Simplified 31.3%

   \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

    2. lower-+.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

    3. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

    4. lower-sin.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

    5. lower-cos.f64 40.8

       \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

11. Simplified 40.8%

    \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]
12. Step-by-step derivation

    1. tan-quot N/A

       \[\leadsto \color{blue}{\tan y} + x \]

    2. lift-tan.f64 40.8

       \[\leadsto \color{blue}{\tan y} + x \]

13. Applied egg-rr 40.8%

    \[\leadsto \color{blue}{\tan y} + x \]

14. Final simplification 40.8%

    \[\leadsto x + \tan y \]
15. Add Preprocessing

Alternative 10: 24.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6200000:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -6200000.0)
   (fma a (fma -0.3333333333333333 (* a a) -1.0) x)
   (+ x y)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -6200000.0) {
		tmp = fma(a, fma(-0.3333333333333333, (a * a), -1.0), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -6200000.0)
		tmp = fma(a, fma(-0.3333333333333333, Float64(a * a), -1.0), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -6200000.0], N[(a * N[(-0.3333333333333333 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e6

   1. Initial program 66.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 38.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 38.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 38.0

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)\right) + x} \]

       3. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot {a}^{3}\right)\right)\right)} + x \]

       4. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{-1 \cdot a} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot {a}^{3}\right)\right)\right) + x \]

       5. distribute-lft-neg-in N/A

          \[\leadsto \left(-1 \cdot a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {a}^{3}}\right) + x \]

       6. metadata-eval N/A

          \[\leadsto \left(-1 \cdot a + \color{blue}{\frac{-1}{3}} \cdot {a}^{3}\right) + x \]

       7. unpow3 N/A

          \[\leadsto \left(-1 \cdot a + \frac{-1}{3} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)}\right) + x \]

       8. unpow2 N/A

          \[\leadsto \left(-1 \cdot a + \frac{-1}{3} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right)\right) + x \]

       9. associate-*r* N/A

          \[\leadsto \left(-1 \cdot a + \color{blue}{\left(\frac{-1}{3} \cdot {a}^{2}\right) \cdot a}\right) + x \]

       10. distribute-rgt-in N/A

           \[\leadsto \color{blue}{a \cdot \left(-1 + \frac{-1}{3} \cdot {a}^{2}\right)} + x \]

       11. +-commutative N/A

           \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{3} \cdot {a}^{2} + -1\right)} + x \]

       12. metadata-eval N/A

           \[\leadsto a \cdot \left(\frac{-1}{3} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) + x \]

       13. sub-neg N/A

           \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{3} \cdot {a}^{2} - 1\right)} + x \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{3} \cdot {a}^{2} - 1, x\right)} \]

       15. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{3} \cdot {a}^{2} + \left(\mathsf{neg}\left(1\right)\right)}, x\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, \frac{-1}{3} \cdot {a}^{2} + \color{blue}{-1}, x\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {a}^{2}, -1\right)}, x\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{a \cdot a}, -1\right), x\right) \]

       19. lower-*.f64 13.8

           \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, \color{blue}{a \cdot a}, -1\right), x\right) \]

   11. Simplified 13.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), x\right)} \]

   if -6.2e6 < y

   1. Initial program 85.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

      6. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

      9. lower-*.f64 41.9

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

   5. Simplified 41.9%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

      10. cube-mult N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      14. lower-*.f64 29.1

          \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   8. Simplified 29.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

       4. lower-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

       5. lower-cos.f64 39.0

          \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

   12. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 29.3

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

   14. Simplified 29.3%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6200000:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 22.1% accurate, 52.5× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x y))

C

double code(double x, double y, double z, double a) {
	return x + y;
}

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + y;
}

Python

def code(x, y, z, a):
	return x + y

Julia

function code(x, y, z, a)
	return Float64(x + y)
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + y;
end

Wolfram

code[x_, y_, z_, a_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, \frac{1}{3} \cdot {a}^{2}, a\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \frac{1}{3}}, a\right)\right) \]

   6. unpow2 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{3}, a\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, \color{blue}{a \cdot \left(a \cdot \frac{1}{3}\right)}, a\right)\right) \]

   9. lower-*.f64 41.0

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a \cdot 0.3333333333333333\right)}, a\right)\right) \]

5. Simplified 41.0%

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + \left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\sin y}{\cos y} + \color{blue}{\left(x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\sin y}{\cos y} + \left(x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)}\right) \]

   10. cube-mult N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   14. lower-*.f64 31.3

       \[\leadsto \frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

8. Simplified 31.3%

   \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + \left(x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

    2. lower-+.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

    3. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} + x \]

    4. lower-sin.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin y}}{\cos y} + x \]

    5. lower-cos.f64 40.8

       \[\leadsto \frac{\sin y}{\color{blue}{\cos y}} + x \]

11. Simplified 40.8%

    \[\leadsto \color{blue}{\frac{\sin y}{\cos y} + x} \]

12. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 22.4

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

14. Simplified 22.4%

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

15. Final simplification 22.4%

    \[\leadsto x + y \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))