tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 32.0s

Alternatives: 15

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 rewrites 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: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}, t\_0, x - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.02)
     t_1
     (if (<= (tan a) 1e-14)
       (fma
        (/ 1.0 (fma (tan y) (- (tan z)) 1.0))
        t_0
        (- x (fma (* a a) (* a 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_1;
	} else if (tan(a) <= 1e-14) {
		tmp = fma((1.0 / fma(tan(y), -tan(z), 1.0)), t_0, (x - fma((a * a), (a * 0.3333333333333333), a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_1;
	elseif (tan(a) <= 1e-14)
		tmp = fma(Float64(1.0 / fma(tan(y), Float64(-tan(z)), 1.0)), t_0, Float64(x - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(N[(1.0 / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. tan-sum N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 81.9%

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

   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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-+l+ N/A

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

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}, \tan y + \tan z, \left(-\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

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

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;x + \left(\frac{t\_0}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.02)
     t_1
     (if (<= (tan a) 1e-14)
       (+
        x
        (-
         (/ t_0 (fma (tan y) (- (tan z)) 1.0))
         (fma (* a a) (* a 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_1;
	} else if (tan(a) <= 1e-14) {
		tmp = x + ((t_0 / fma(tan(y), -tan(z), 1.0)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_1;
	elseif (tan(a) <= 1e-14)
		tmp = Float64(x + Float64(Float64(t_0 / fma(tan(y), Float64(-tan(z)), 1.0)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(x + N[(N[(t$95$0 / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. tan-sum N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 81.9%

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

   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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

      1. tan-sum N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 99.9

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 10^{-14}:\\ \;\;\;\;\frac{t\_0}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.02)
     t_1
     (if (<= (tan a) 1e-14)
       (+ (/ t_0 (- 1.0 (* (tan y) (tan z)))) (- x a))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_1;
	} else if (tan(a) <= 1e-14) {
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) + (x - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_1;
	elseif (tan(a) <= 1e-14)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-14], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. tan-sum N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 81.9%

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

   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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.0

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

   8. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + \left(x - a\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. quot-tan N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. tan-sum N/A

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

      7. tan-quot N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. un-div-inv N/A

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

      18. lift-/.f64 N/A

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

      19. lift-tan.f64 N/A

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

      20. unsub-neg N/A

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

   10. Applied rewrites 99.9%

       \[\leadsto \color{blue}{\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y}} + \left(x - a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

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

Alternative 5: 80.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(1.0 * N[(N[Tan[y], $MachinePrecision] + N[Tan[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. 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 rewrites 99.8%

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

   1. tan-sum N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift--.f64 N/A

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

   16. clear-num N/A

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

   17. associate-/r/ N/A

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

   18. lower-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in y around 0

   \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 81.1%

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

Alternative 6: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan z - \tan a\right)\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan z) (tan a)))))
   (if (<= a -1.9e-8)
     t_0
     (if (<= a 0.6)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma
           a
           (* a (fma (* a a) 0.05396825396825397 0.13333333333333333))
           0.3333333333333333)
          (* a (* a a))
          a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(z) - tan(a));
	double tmp;
	if (a <= -1.9e-8) {
		tmp = t_0;
	} else if (a <= 0.6) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * fma((a * a), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(z) - tan(a)))
	tmp = 0.0
	if (a <= -1.9e-8)
		tmp = t_0;
	elseif (a <= 0.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * fma(Float64(a * a), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.9e-8], t$95$0, If[LessEqual[a, 0.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * N[(N[(a * a), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.90000000000000014e-8 or 0.599999999999999978 < a

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 59.6

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

   5. Applied rewrites 59.6%

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.6

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 59.6

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

   7. Applied rewrites 59.6%

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

   if -1.90000000000000014e-8 < a < 0.599999999999999978

   1. Initial program 80.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right) \]
   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. Applied rewrites 80.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{elif}\;a \leq 0.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8: 55.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;a \leq -520000000000:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z \cdot z, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), t\_0, z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.4:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, t\_0, z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= a -520000000000.0)
     (+
      x
      (-
       (fma
        (fma
         (* z z)
         (fma (* z z) 0.05396825396825397 0.13333333333333333)
         0.3333333333333333)
        t_0
        z)
       (tan a)))
     (if (<= a 1.4)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma
           a
           (* a (fma (* a a) 0.05396825396825397 0.13333333333333333))
           0.3333333333333333)
          (* a (* a a))
          a)))
       (+ x (- (fma 0.3333333333333333 t_0 z) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = z * (z * z);
	double tmp;
	if (a <= -520000000000.0) {
		tmp = x + (fma(fma((z * z), fma((z * z), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), t_0, z) - tan(a));
	} else if (a <= 1.4) {
		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 + (fma(0.3333333333333333, t_0, z) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (a <= -520000000000.0)
		tmp = Float64(x + Float64(fma(fma(Float64(z * z), fma(Float64(z * z), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), t_0, z) - tan(a)));
	elseif (a <= 1.4)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * fma(Float64(a * a), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(fma(0.3333333333333333, t_0, z) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -520000000000.0], N[(x + N[(N[(N[(N[(z * z), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * N[(N[(a * a), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2e11

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.2%

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

   if -5.2e11 < a < 1.3999999999999999

   1. Initial program 80.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Applied rewrites 78.7%

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

   if 1.3999999999999999 < a

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right)} - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;a \leq -520000000000:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z \cdot z, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), t\_0, z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.3:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, t\_0, z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= a -520000000000.0)
     (+
      x
      (-
       (fma
        (fma
         (* z z)
         (fma (* z z) 0.05396825396825397 0.13333333333333333)
         0.3333333333333333)
        t_0
        z)
       (tan a)))
     (if (<= a 1.3)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 0.13333333333333333) 0.3333333333333333)
          (* a (* a a))
          a)))
       (+ x (- (fma 0.3333333333333333 t_0 z) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = z * (z * z);
	double tmp;
	if (a <= -520000000000.0) {
		tmp = x + (fma(fma((z * z), fma((z * z), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), t_0, z) - tan(a));
	} else if (a <= 1.3) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = x + (fma(0.3333333333333333, t_0, z) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (a <= -520000000000.0)
		tmp = Float64(x + Float64(fma(fma(Float64(z * z), fma(Float64(z * z), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), t_0, z) - tan(a)));
	elseif (a <= 1.3)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(fma(0.3333333333333333, t_0, z) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -520000000000.0], N[(x + N[(N[(N[(N[(z * z), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2e11

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.2%

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

   if -5.2e11 < a < 1.30000000000000004

   1. Initial program 80.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 1.30000000000000004 < a

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right)} - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;a \leq -520000000000:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, z \cdot z, 0.3333333333333333\right), t\_0, z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.3:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, t\_0, z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= a -520000000000.0)
     (+
      x
      (-
       (fma (fma 0.13333333333333333 (* z z) 0.3333333333333333) t_0 z)
       (tan a)))
     (if (<= a 1.3)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 0.13333333333333333) 0.3333333333333333)
          (* a (* a a))
          a)))
       (+ x (- (fma 0.3333333333333333 t_0 z) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = z * (z * z);
	double tmp;
	if (a <= -520000000000.0) {
		tmp = x + (fma(fma(0.13333333333333333, (z * z), 0.3333333333333333), t_0, z) - tan(a));
	} else if (a <= 1.3) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = x + (fma(0.3333333333333333, t_0, z) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (a <= -520000000000.0)
		tmp = Float64(x + Float64(fma(fma(0.13333333333333333, Float64(z * z), 0.3333333333333333), t_0, z) - tan(a)));
	elseif (a <= 1.3)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = Float64(x + Float64(fma(0.3333333333333333, t_0, z) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -520000000000.0], N[(x + N[(N[(N[(0.13333333333333333 * N[(z * z), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2e11

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. cube-unmult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 29.2

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

   8. Applied rewrites 29.2%

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

   if -5.2e11 < a < 1.30000000000000004

   1. Initial program 80.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 1.30000000000000004 < a

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right)} - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ \mathbf{if}\;a \leq -520000000000:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, z \cdot z, 0.3333333333333333\right), t\_0, z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(0.3333333333333333, t\_0, z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* z (* z z))))
   (if (<= a -520000000000.0)
     (+
      x
      (-
       (fma (fma 0.13333333333333333 (* z z) 0.3333333333333333) t_0 z)
       (tan a)))
     (if (<= a 1.4e+17)
       (- (+ x (tan (+ y z))) a)
       (+ x (- (fma 0.3333333333333333 t_0 z) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = z * (z * z);
	double tmp;
	if (a <= -520000000000.0) {
		tmp = x + (fma(fma(0.13333333333333333, (z * z), 0.3333333333333333), t_0, z) - tan(a));
	} else if (a <= 1.4e+17) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = x + (fma(0.3333333333333333, t_0, z) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(z * Float64(z * z))
	tmp = 0.0
	if (a <= -520000000000.0)
		tmp = Float64(x + Float64(fma(fma(0.13333333333333333, Float64(z * z), 0.3333333333333333), t_0, z) - tan(a)));
	elseif (a <= 1.4e+17)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = Float64(x + Float64(fma(0.3333333333333333, t_0, z) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -520000000000.0], N[(x + N[(N[(N[(0.13333333333333333 * N[(z * z), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e+17], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 * t$95$0 + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2e11

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. cube-unmult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 29.2

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

   8. Applied rewrites 29.2%

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

   if -5.2e11 < a < 1.4e17

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 76.8

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

   8. Applied rewrites 76.8%

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

   10. Applied rewrites 76.8%

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

   if 1.4e17 < a

   1. Initial program 83.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 37.2

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

   8. Applied rewrites 37.2%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right)} - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\mathsf{fma}\left(0.3333333333333333, z \cdot \left(z \cdot z\right), z\right) - \tan a\right)\\ \mathbf{if}\;a \leq -520000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (fma 0.3333333333333333 (* z (* z z)) z) (tan a)))))
   (if (<= a -520000000000.0)
     t_0
     (if (<= a 1.4e+17) (- (+ x (tan (+ y z))) a) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (fma(0.3333333333333333, (z * (z * z)), z) - tan(a));
	double tmp;
	if (a <= -520000000000.0) {
		tmp = t_0;
	} else if (a <= 1.4e+17) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(fma(0.3333333333333333, Float64(z * Float64(z * z)), z) - tan(a)))
	tmp = 0.0
	if (a <= -520000000000.0)
		tmp = t_0;
	elseif (a <= 1.4e+17)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(0.3333333333333333 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -520000000000.0], t$95$0, If[LessEqual[a, 1.4e+17], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.2e11 or 1.4e17 < a

   1. Initial program 81.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin z}{\cos z}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.7

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

   8. Applied rewrites 33.7%

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

   if -5.2e11 < a < 1.4e17

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 76.8

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

   8. Applied rewrites 76.8%

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

   10. Applied rewrites 76.8%

       \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 65.9

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

5. Applied rewrites 65.9%

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

6. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 41.5

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

8. Applied rewrites 41.5%

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

10. Applied rewrites 41.4%

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

Alternative 14: 21.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ x + a \cdot \left(a \cdot \left(a \cdot -0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (* a (* a (* a -0.3333333333333333)))))

C

double code(double x, double y, double z, double a) {
	return x + (a * (a * (a * -0.3333333333333333)));
}

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 + (a * (a * (a * (-0.3333333333333333d0))))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (a * (a * (a * -0.3333333333333333)));
}

Python

def code(x, y, z, a):
	return x + (a * (a * (a * -0.3333333333333333)))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(a * Float64(a * Float64(a * -0.3333333333333333))))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (a * (a * (a * -0.3333333333333333)));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(a * N[(a * N[(a * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 41.0

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

5. Applied rewrites 41.0%

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

6. Taylor expanded in a around inf

   \[\leadsto x + \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 22.0

       \[\leadsto x + a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.3333333333333333\right)}\right) \]

8. Applied rewrites 22.0%

   \[\leadsto x + \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.3333333333333333\right)\right)} \]
9. Add Preprocessing

Alternative 15: 3.5% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double a) {
	return -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 = -a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := (-a)

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 65.9

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

5. Applied rewrites 65.9%

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

6. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 41.5

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

8. Applied rewrites 41.5%

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

9. Taylor expanded in a around inf

   \[\leadsto \color{blue}{-1 \cdot a} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

    2. lower-neg.f64 3.4

       \[\leadsto \color{blue}{-a} \]

11. Applied rewrites 3.4%

    \[\leadsto \color{blue}{-a} \]
12. 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))))