tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 32.0s

Alternatives: 9

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: 79.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-neg.f64 N/A

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

   7. lower-fma.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ t_1 := x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 - \tan y, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))) (t_1 (- x (+ (* -1.0 (+ (tan z) (tan y))) (tan a)))))
   (if (<= (tan a) -0.002)
     t_1
     (if (<= (tan a) 2e-9)
       (fma (- t_0 (tan y)) (/ -1.0 (fma t_0 (tan y) 1.0)) (- x a))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double t_1 = x - ((-1.0 * (tan(z) + tan(y))) + tan(a));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = t_1;
	} else if (tan(a) <= 2e-9) {
		tmp = fma((t_0 - tan(y)), (-1.0 / fma(t_0, tan(y), 1.0)), (x - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	t_1 = Float64(x - Float64(Float64(-1.0 * Float64(tan(z) + tan(y))) + tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = t_1;
	elseif (tan(a) <= 2e-9)
		tmp = fma(Float64(t_0 - tan(y)), Float64(-1.0 / fma(t_0, tan(y), 1.0)), Float64(x - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, Block[{t$95$1 = N[(x - N[(N[(-1.0 * N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-9], N[(N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 2.00000000000000012e-9 < (tan.f64 a)

   1. Initial program 83.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.5%

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

   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 83.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.9

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

      9. lift-neg.f64 N/A

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

      10. lift-+.f64 N/A

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

   9. Applied rewrites 83.9%

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

   if -2e-3 < (tan.f64 a) < 2.00000000000000012e-9

   1. Initial program 80.5%

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

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (* -1.0 (+ (tan z) (tan y))) (tan a)))))
   (if (<= (tan a) -0.002)
     t_0
     (if (<= (tan a) 2e-12)
       (- (/ (- (- (tan z)) (tan y)) (fma (tan z) (tan y) -1.0)) (- a x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - ((-1.0 * (tan(z) + tan(y))) + tan(a));
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = t_0;
	} else if (tan(a) <= 2e-12) {
		tmp = ((-tan(z) - tan(y)) / fma(tan(z), tan(y), -1.0)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(-1.0 * Float64(tan(z) + tan(y))) + tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = t_0;
	elseif (tan(a) <= 2e-12)
		tmp = Float64(Float64(Float64(Float64(-tan(z)) - tan(y)) / fma(tan(z), tan(y), -1.0)) - Float64(a - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(-1.0 * N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-12], 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[(a - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -2e-3 or 1.99999999999999996e-12 < (tan.f64 a)

   1. Initial program 83.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. 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 99.5%

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

   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 83.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.0

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

      9. lift-neg.f64 N/A

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

      10. lift-+.f64 N/A

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

   9. Applied rewrites 84.0%

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

   if -2e-3 < (tan.f64 a) < 1.99999999999999996e-12

   1. Initial program 80.4%

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

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

   4. Applied rewrites 80.4%

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

   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 80.4

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

   7. Applied rewrites 80.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-/.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (+ (* -1.0 (+ (tan z) (tan y))) (tan a)))))
   (if (<= (tan a) -5e-7)
     t_0
     (if (<= (tan a) 2e-12)
       (- (/ (- (- (tan z)) (tan y)) (fma (tan z) (tan y) -1.0)) (- x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - ((-1.0 * (tan(z) + tan(y))) + tan(a));
	double tmp;
	if (tan(a) <= -5e-7) {
		tmp = t_0;
	} else if (tan(a) <= 2e-12) {
		tmp = ((-tan(z) - tan(y)) / fma(tan(z), tan(y), -1.0)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - Float64(Float64(-1.0 * Float64(tan(z) + tan(y))) + tan(a)))
	tmp = 0.0
	if (tan(a) <= -5e-7)
		tmp = t_0;
	elseif (tan(a) <= 2e-12)
		tmp = Float64(Float64(Float64(Float64(-tan(z)) - tan(y)) / fma(tan(z), tan(y), -1.0)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[(N[(-1.0 * N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-7], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-12], 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] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (tan.f64 a)

   1. Initial program 83.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.5%

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

   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 84.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-neg.f64 N/A

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

      10. lift-+.f64 N/A

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

   9. Applied rewrites 84.1%

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

   if -4.99999999999999977e-7 < (tan.f64 a) < 1.99999999999999996e-12

   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. +-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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 79.9

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

   7. Applied rewrites 79.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lift-neg.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-tan.f64 N/A

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

      16. lift-tan.f64 N/A

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

   9. Applied rewrites 99.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-7}:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

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

Alternative 6: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x - ((-1.0 * (tan(z) + 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 - (((-1.0d0) * (tan(z) + tan(y))) + tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 82.6

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

   9. lift-neg.f64 N/A

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

   10. lift-+.f64 N/A

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

9. Applied rewrites 82.6%

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

10. Final simplification 82.6%

    \[\leadsto x - \left(-1 \cdot \left(\tan z + \tan y\right) + \tan a\right) \]
11. Add Preprocessing

Alternative 7: 79.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.6%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

9. Applied rewrites 82.6%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Final simplification 82.1%

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

Alternative 9: 51.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.0

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

4. Applied rewrites 82.0%

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

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 51.5

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

7. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2024243 
(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))))