tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 31.7s

Alternatives: 16

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: 79.1% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan y \cdot \tan z\\ \mathsf{fma}\left(\cos a \cdot \left(\tan y + \tan z\right) - \sin a \cdot t\_0, \frac{1}{\cos a \cdot t\_0}, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan y) (tan z)))))
   (fma
    (- (* (cos a) (+ (tan y) (tan z))) (* (sin a) t_0))
    (/ 1.0 (* (cos a) t_0))
    x)))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return fma(((cos(a) * (tan(y) + tan(z))) - (sin(a) * t_0)), (1.0 / (cos(a) * t_0)), x);
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 - Float64(tan(y) * tan(z)))
	return fma(Float64(Float64(cos(a) * Float64(tan(y) + tan(z))) - Float64(sin(a) * t_0)), Float64(1.0 / Float64(cos(a) * t_0)), x)
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Cos[a], $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Cos[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Initial program 82.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(\frac{t\_0}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(-0.3333333333333333, y \cdot y, 1\right)}{y}}} - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, x - \mathsf{fma}\left(a, 0.3333333333333333 \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z)))
        (t_1
         (+
          x
          (-
           (/
            t_0
            (- 1.0 (/ (tan z) (/ (fma -0.3333333333333333 (* y y) 1.0) y))))
           (tan a)))))
   (if (<= (tan a) -0.05)
     t_1
     (if (<= (tan a) 2.6e-14)
       (fma
        (/ 1.0 (- 1.0 (* (tan y) (tan z))))
        t_0
        (- x (fma a (* 0.3333333333333333 (* a a)) 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 + ((t_0 / (1.0 - (tan(z) / (fma(-0.3333333333333333, (y * y), 1.0) / y)))) - tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_1;
	} else if (tan(a) <= 2.6e-14) {
		tmp = fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, (x - fma(a, (0.3333333333333333 * (a * a)), 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 + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) / Float64(fma(-0.3333333333333333, Float64(y * y), 1.0) / y)))) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_1;
	elseif (tan(a) <= 2.6e-14)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, Float64(x - fma(a, Float64(0.3333333333333333 * Float64(a * a)), 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[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] / N[(N[(-0.3333333333333333 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 2.6e-14], N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - N[(a * N[(0.3333333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 2.59999999999999997e-14 < (tan.f64 a)

   1. Initial program 83.4%

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

      1. lift-tan.f64 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. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-tan.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. lower-tan.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.1

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

   9. Applied rewrites 84.1%

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

   if -0.050000000000000003 < (tan.f64 a) < 2.59999999999999997e-14

   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 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. lift-tan.f64 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. tan-quot N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-+.f64 N/A

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - ((tan(y) * sin(z)) / cos(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) * sin(z)) / cos(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.sin(z)) / Math.cos(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.sin(z)) / math.cos(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(Float64(tan(y) * sin(z)) / cos(z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - ((tan(y) * sin(z)) / cos(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[(N[Tan[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-tan.f64 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. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. 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. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\tan y}\\ x + \left(\frac{\mathsf{fma}\left(-1, \frac{1}{t\_0}, \tan z\right)}{1 + \frac{\tan z}{t\_0}} - \tan a\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ -1.0 (tan y))))
   (+
    x
    (- (/ (fma -1.0 (/ 1.0 t_0) (tan z)) (+ 1.0 (/ (tan z) t_0))) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -1.0 / tan(y);
	return x + ((fma(-1.0, (1.0 / t_0), tan(z)) / (1.0 + (tan(z) / t_0))) - tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-1.0 / tan(y))
	return Float64(x + Float64(Float64(fma(-1.0, Float64(1.0 / t_0), tan(z)) / Float64(1.0 + Float64(tan(z) / t_0))) - tan(a)))
end

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. lift-tan.f64 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. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. 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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. clear-num N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f64 N/A

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

   4. inv-pow N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-/.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. metadata-eval N/A

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

   13. inv-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(-1.0 * N[(1.0 / N[(-1.0 / N[Tan[y], $MachinePrecision]), $MachinePrecision]), $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 82.0%

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

   1. lift-tan.f64 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. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. 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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. clear-num N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f64 N/A

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

   4. inv-pow N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-/.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. metadata-eval N/A

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

   13. inv-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lower-*.f64 99.6

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 6: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 + (tan(z) / (-1.0 / tan(y))))) - 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(z) / ((-1.0d0) / tan(y))))) - 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(z) / (-1.0 / Math.tan(y))))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 + (math.tan(z) / (-1.0 / math.tan(y))))) - 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(z) / Float64(-1.0 / tan(y))))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 + (tan(z) / (-1.0 / tan(y))))) - 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[z], $MachinePrecision] / N[(-1.0 / N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-tan.f64 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. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. 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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. clear-num N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 7: 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 82.0%

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

   1. lift-tan.f64 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. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. 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. Add Preprocessing

Alternative 8: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0102:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, x - \mathsf{fma}\left(a, 0.3333333333333333 \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.0102)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= a 2.7e-14)
     (fma
      (/ 1.0 (- 1.0 (* (tan y) (tan z))))
      (+ (tan y) (tan z))
      (- x (fma a (* 0.3333333333333333 (* a a)) a)))
     (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.0102) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 2.7e-14) {
		tmp = fma((1.0 / (1.0 - (tan(y) * tan(z)))), (tan(y) + tan(z)), (x - fma(a, (0.3333333333333333 * (a * a)), a)));
	} else {
		tmp = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.0102)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 2.7e-14)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), Float64(tan(y) + tan(z)), Float64(x - fma(a, Float64(0.3333333333333333 * Float64(a * a)), a)));
	else
		tmp = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.0102], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.7e-14], N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + N[(x - N[(a * N[(0.3333333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.010200000000000001

   1. Initial program 81.9%

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

   if -0.010200000000000001 < a < 2.6999999999999999e-14

   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 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. lift-tan.f64 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. tan-quot N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-+.f64 N/A

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 99.7%

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

   if 2.6999999999999999e-14 < a

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-neg.f64 85.2

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

   4. Applied rewrites 85.2%

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

Alternative 9: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0102:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{\cos \left(y + z\right)}, \sin \left(y + z\right), -\tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.0102)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= a 2.7e-14)
     (+
      x
      (-
       (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))
       (fma (* a a) (* a 0.3333333333333333) a)))
     (+ x (fma (/ 1.0 (cos (+ y z))) (sin (+ y z)) (- (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.0102) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 2.7e-14) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + fma((1.0 / cos((y + z))), sin((y + z)), -tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.0102)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 2.7e-14)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + fma(Float64(1.0 / cos(Float64(y + z))), sin(Float64(y + z)), Float64(-tan(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.0102], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.7e-14], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.010200000000000001

   1. Initial program 81.9%

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

   if -0.010200000000000001 < a < 2.6999999999999999e-14

   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 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 2.6999999999999999e-14 < a

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-neg.f64 85.2

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

   4. Applied rewrites 85.2%

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

Alternative 10: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 5e-14)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 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(y) - tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 5e-14) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 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(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 5e-14)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-14], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 5.0000000000000002e-14 < (tan.f64 a)

   1. Initial program 83.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 63.0

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

   5. Applied rewrites 63.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if -0.050000000000000003 < (tan.f64 a) < 5.0000000000000002e-14

   1. Initial program 80.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0002:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;\tan a \leq 0.08:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.0002)
   (/ 1.0 (/ 1.0 x))
   (if (<= (tan a) 0.08)
     (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
     (+ x (- (fma y (* (* y y) 0.3333333333333333) y) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.0002) {
		tmp = 1.0 / (1.0 / x);
	} else if (tan(a) <= 0.08) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + (fma(y, ((y * y) * 0.3333333333333333), y) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.0002)
		tmp = Float64(1.0 / Float64(1.0 / x));
	elseif (tan(a) <= 0.08)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(fma(y, Float64(Float64(y * y) * 0.3333333333333333), y) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.0002], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.08], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2.0000000000000001e-4

   1. Initial program 78.5%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 78.3

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 22.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -2.0000000000000001e-4 < (tan.f64 a) < 0.0800000000000000017

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 80.4

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

   5. Applied rewrites 80.4%

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

   if 0.0800000000000000017 < (tan.f64 a)

   1. Initial program 85.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0002:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;\tan a \leq 0.08:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.1% 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 82.0%

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

Alternative 13: 54.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.13333333333333333, 0.3333333333333333\right), y \cdot \left(y \cdot y\right), y\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 6.9 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.05e+22)
   (+
    x
    (-
     (fma (fma (* y y) 0.13333333333333333 0.3333333333333333) (* y (* y y)) y)
     (tan a)))
   (if (<= a 6.9e-12)
     (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
     (+ x (- (fma y (* (* y y) 0.3333333333333333) y) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.05e+22) {
		tmp = x + (fma(fma((y * y), 0.13333333333333333, 0.3333333333333333), (y * (y * y)), y) - tan(a));
	} else if (a <= 6.9e-12) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + (fma(y, ((y * y) * 0.3333333333333333), y) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.05e+22)
		tmp = Float64(x + Float64(fma(fma(Float64(y * y), 0.13333333333333333, 0.3333333333333333), Float64(y * Float64(y * y)), y) - tan(a)));
	elseif (a <= 6.9e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(fma(y, Float64(Float64(y * y) * 0.3333333333333333), y) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.05e+22], N[(x + N[(N[(N[(N[(y * y), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.9e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0499999999999999e22

   1. Initial program 84.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.5%

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

   if -1.0499999999999999e22 < a < 6.9000000000000001e-12

   1. Initial program 79.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. 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 78.6

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

   5. Applied rewrites 78.6%

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

   if 6.9000000000000001e-12 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.13333333333333333, 0.3333333333333333\right), y \cdot \left(y \cdot y\right), y\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 6.9 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -1.45)
   (/ 1.0 (/ 1.0 x))
   (+ x (- (fma y (* (* y y) 0.3333333333333333) y) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -1.45) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = x + (fma(y, ((y * y) * 0.3333333333333333), y) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -1.45)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(x + Float64(fma(y, Float64(Float64(y * y) * 0.3333333333333333), y) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -1.45], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + y), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 65.6

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 24.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 24.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.44999999999999996 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 43.1%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.3333333333333333, y\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

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 = 1.0d0 / (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Python

def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

function code(x, y, z, a)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 81.9

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

4. Applied rewrites 81.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.4

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Add Preprocessing

Alternative 16: 2.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-1}{x}} \end{array} \]

FPCore

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

C

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

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 = 1.0d0 / ((-1.0d0) / x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 / (-1.0 / x);
}

Python

def code(x, y, z, a):
	return 1.0 / (-1.0 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 81.9

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

4. Applied rewrites 81.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.4

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Step-by-step derivation

9. Applied rewrites 32.4%

   \[\leadsto \frac{1}{{\left(x \cdot x\right)}^{\color{blue}{-0.5}}} \]

10. Taylor expanded in x around -inf

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{x}}} \]
11. Step-by-step derivation

12. Applied rewrites 3.0%

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{x}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(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))))