tan-example (used to crash)

Percentage Accurate: 78.7% → 99.7%

Time: 30.9s

Alternatives: 14

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.7165776239542025
  2. y = 1.4427356365040286e+208
  3. z = 2.8160000590559827e+208
  4. a = 1.1316928951422652e-86

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: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  (-
   (/ (fma (sin z) (/ 1.0 (cos z)) (tan y)) (fma (- (tan z)) (tan y) 1.0))
   (tan a))
  x))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[(1.0 / N[Cos[z], $MachinePrecision]), $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := -\tan z\\ t_1 := t\_0 - \tan y\\ \mathbf{if}\;\tan a \leq -0.0002:\\ \;\;\;\;\left(\tan \left(y + z\right) - \frac{1}{\frac{1}{\tan a}}\right) + x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))) (t_1 (- t_0 (tan y))))
   (if (<= (tan a) -0.0002)
     (+ (- (tan (+ y z)) (/ 1.0 (/ 1.0 (tan a)))) x)
     (if (<= (tan a) 5e-43)
       (fma t_1 (/ -1.0 (fma t_0 (tan y) 1.0)) (- x a))
       (fma t_1 -1.0 (- x (tan a)))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double t_1 = t_0 - tan(y);
	double tmp;
	if (tan(a) <= -0.0002) {
		tmp = (tan((y + z)) - (1.0 / (1.0 / tan(a)))) + x;
	} else if (tan(a) <= 5e-43) {
		tmp = fma(t_1, (-1.0 / fma(t_0, tan(y), 1.0)), (x - a));
	} else {
		tmp = fma(t_1, -1.0, (x - tan(a)));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	t_1 = Float64(t_0 - tan(y))
	tmp = 0.0
	if (tan(a) <= -0.0002)
		tmp = Float64(Float64(tan(Float64(y + z)) - Float64(1.0 / Float64(1.0 / tan(a)))) + x);
	elseif (tan(a) <= 5e-43)
		tmp = fma(t_1, Float64(-1.0 / fma(t_0, tan(y), 1.0)), Float64(x - a));
	else
		tmp = fma(t_1, -1.0, Float64(x - tan(a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, Block[{t$95$1 = N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.0002], N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(1.0 / N[(1.0 / N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-43], N[(t$95$1 * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * -1.0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 81.5%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if -2.0000000000000001e-4 < (tan.f64 a) < 5.00000000000000019e-43

   1. Initial program 76.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 5.00000000000000019e-43 < (tan.f64 a)

   1. Initial program 85.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 86.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0002:\\ \;\;\;\;\left(\tan \left(y + z\right) - \frac{1}{\frac{1}{\tan a}}\right) + x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, -1, x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ (- (/ (+ (tan z) (tan y)) (fma (- (tan z)) (tan y) 1.0)) (tan a)) x))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 58.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;\tan a \leq -0.175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= (tan a) -0.175)
     t_0
     (if (<= (tan a) 2e-13) (- (tan (+ y z)) (- x)) t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (tan(a) <= -0.175) {
		tmp = t_0;
	} else if (tan(a) <= 2e-13) {
		tmp = tan((y + z)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (tan(a) <= (-0.175d0)) then
        tmp = t_0
    else if (tan(a) <= 2d-13) then
        tmp = tan((y + z)) - -x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (Math.tan(a) <= -0.175) {
		tmp = t_0;
	} else if (Math.tan(a) <= 2e-13) {
		tmp = Math.tan((y + z)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if math.tan(a) <= -0.175:
		tmp = t_0
	elif math.tan(a) <= 2e-13:
		tmp = math.tan((y + z)) - -x
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (tan(a) <= -0.175)
		tmp = t_0;
	elseif (tan(a) <= 2e-13)
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (tan(a) <= -0.175)
		tmp = t_0;
	elseif (tan(a) <= 2e-13)
		tmp = tan((y + z)) - -x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.175], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-13], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.17499999999999999 or 2.0000000000000001e-13 < (tan.f64 a)

   1. Initial program 82.0%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.9%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.9%

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

   if -0.17499999999999999 < (tan.f64 a) < 2.0000000000000001e-13

   1. Initial program 78.4%

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

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.0

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

   7. Applied rewrites 76.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.175:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma (- (- (tan z)) (tan y)) -1.0 (- x (tan a))))

C

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return fma(Float64(Float64(-tan(z)) - tan(y)), -1.0, Float64(x - tan(a)))
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[((-N[Tan[z], $MachinePrecision]) - N[Tan[y], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 80.7%

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

8. Final simplification 80.7%

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

Alternative 6: 78.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ (- (tan (+ y z)) (/ 1.0 (/ 1.0 (tan a)))) x))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 = (tan((y + z)) - (1.0d0 / (1.0d0 / tan(a)))) + x
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return (math.tan((y + z)) - (1.0 / (1.0 / math.tan(a)))) + x

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(Float64(tan(Float64(y + z)) - Float64(1.0 / Float64(1.0 / tan(a)))) + x)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (tan((y + z)) - (1.0 / (1.0 / tan(a)))) + x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(1.0 / N[(1.0 / N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 80.2

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

4. Applied rewrites 80.2%

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

5. Final simplification 80.2%

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

Alternative 7: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-12}:\\ \;\;\;\;\left(\tan y - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-12) (+ (- (tan y) (tan a)) x) (- (tan (+ y z)) (- x))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-12) {
		tmp = (tan(y) - tan(a)) + x;
	} else {
		tmp = tan((y + z)) - -x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if ((y + z) <= 1d-12) then
        tmp = (tan(y) - tan(a)) + x
    else
        tmp = tan((y + z)) - -x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-12) {
		tmp = (Math.tan(y) - Math.tan(a)) + x;
	} else {
		tmp = Math.tan((y + z)) - -x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 1e-12:
		tmp = (math.tan(y) - math.tan(a)) + x
	else:
		tmp = math.tan((y + z)) - -x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 1e-12)
		tmp = Float64(Float64(tan(y) - tan(a)) + x);
	else
		tmp = Float64(tan(Float64(y + z)) - Float64(-x));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 1e-12)
		tmp = (tan(y) - tan(a)) + x;
	else
		tmp = tan((y + z)) - -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 1e-12], N[(N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 9.9999999999999998e-13

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 65.7%

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

   if 9.9999999999999998e-13 < (+.f64 y z)

   1. Initial program 73.0%

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

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.4

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

   7. Applied rewrites 49.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-12}:\\ \;\;\;\;\left(\tan y - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ (- (tan (+ y z)) (tan a)) x))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 = (tan((y + z)) - tan(a)) + x
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(Float64(tan(Float64(y + z)) - tan(a)) + x)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (tan((y + z)) - tan(a)) + x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Final simplification 80.2%

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

Alternative 9: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.085:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 17:\\ \;\;\;\;\tan \left(y + z\right) - \left(\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.085)
     t_0
     (if (<= a 17.0)
       (- (tan (+ y z)) (- (* (fma 0.3333333333333333 (* a a) 1.0) a) x))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.085) {
		tmp = t_0;
	} else if (a <= 17.0) {
		tmp = tan((y + z)) - ((fma(0.3333333333333333, (a * a), 1.0) * a) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.085)
		tmp = t_0;
	elseif (a <= 17.0)
		tmp = Float64(tan(Float64(y + z)) - Float64(Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a) - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.085], t$95$0, If[LessEqual[a, 17.0], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0850000000000000061 or 17 < a

   1. Initial program 82.1%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.4%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.4%

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

   if -0.0850000000000000061 < a < 17

   1. Initial program 78.2%

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

      1. lift-+.f64 N/A

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

      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. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 8.4

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.5

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

   10. Applied rewrites 77.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.085:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 17:\\ \;\;\;\;\tan \left(y + z\right) - \left(\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 17:\\ \;\;\;\;\tan \left(y + z\right) - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.035) t_0 (if (<= a 17.0) (- (tan (+ y z)) (- a x)) t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.035) {
		tmp = t_0;
	} else if (a <= 17.0) {
		tmp = tan((y + z)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.035d0)) then
        tmp = t_0
    else if (a <= 17.0d0) then
        tmp = tan((y + z)) - (a - x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.035) {
		tmp = t_0;
	} else if (a <= 17.0) {
		tmp = Math.tan((y + z)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.035:
		tmp = t_0
	elif a <= 17.0:
		tmp = math.tan((y + z)) - (a - x)
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.035)
		tmp = t_0;
	elseif (a <= 17.0)
		tmp = Float64(tan(Float64(y + z)) - Float64(a - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.035)
		tmp = t_0;
	elseif (a <= 17.0)
		tmp = tan((y + z)) - (a - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.035], t$95$0, If[LessEqual[a, 17.0], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.035000000000000003 or 17 < a

   1. Initial program 82.1%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.4%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.4%

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

   if -0.035000000000000003 < a < 17

   1. Initial program 78.2%

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

      1. lift-+.f64 N/A

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

      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. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 77.4

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

   7. Applied rewrites 77.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.035:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 17:\\ \;\;\;\;\tan \left(y + z\right) - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x - \tan a \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- x (tan a)))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return x - tan(a);
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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(a)
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x - math.tan(a)

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x - tan(a))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x - tan(a);
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.4

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

5. Applied rewrites 60.4%

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

6. Taylor expanded in y around 0

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

8. Applied rewrites 39.4%

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
9. Step-by-step derivation

10. Applied rewrites 39.4%

    \[\leadsto \color{blue}{x - \tan a} \]
11. Add Preprocessing

Alternative 12: 31.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. lift-+.f64 N/A

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

   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 80.0

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

4. Applied rewrites 79.9%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 29.6

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

7. Applied rewrites 29.6%

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

Alternative 13: 22.5% accurate, 52.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x - a \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- x a))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return x - a;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x - a
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x - a

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x - a)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x - a;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.4

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

5. Applied rewrites 60.4%

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

6. Taylor expanded in y around 0

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

8. Applied rewrites 39.4%

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

9. Taylor expanded in a around 0

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

11. Applied rewrites 20.5%

    \[\leadsto x - a \]
12. Add Preprocessing

Alternative 14: 3.5% accurate, 70.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ -a \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- a))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return -a;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = -a
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return -a

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(-a)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = -a;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := (-a)

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.4

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

5. Applied rewrites 60.4%

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

6. Taylor expanded in y around 0

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

8. Applied rewrites 39.4%

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

9. Taylor expanded in a around 0

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

11. Applied rewrites 20.5%

    \[\leadsto x - a \]

12. Taylor expanded in a around inf

    \[\leadsto -1 \cdot a \]
13. Step-by-step derivation

14. Applied rewrites 3.6%

    \[\leadsto -a \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))