tan-example (used to crash)

Percentage Accurate: 80.0% → 99.7%

Time: 28.1s

Alternatives: 6

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: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;\left(\tan \left(z + y\right) + x\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-54}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{fma}\left(y, \frac{y}{y - z}, \left(-z\right) \cdot \frac{z}{y - z}\right)\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.002)
   (- (+ (tan (+ z y)) x) (tan a))
   (if (<= (tan a) 1e-54)
     (+
      x
      (-
       (/ (+ (tan z) (tan y)) (fma (- (tan z)) (tan y) 1.0))
       (* (fma (* a a) 0.3333333333333333 1.0) a)))
     (+ x (- (tan (fma y (/ y (- y z)) (* (- z) (/ z (- y z))))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = (tan((z + y)) + x) - tan(a);
	} else if (tan(a) <= 1e-54) {
		tmp = x + (((tan(z) + tan(y)) / fma(-tan(z), tan(y), 1.0)) - (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = x + (tan(fma(y, (y / (y - z)), (-z * (z / (y - z))))) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(Float64(tan(Float64(z + y)) + x) - tan(a));
	elseif (tan(a) <= 1e-54)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = Float64(x + Float64(tan(fma(y, Float64(y / Float64(y - z)), Float64(Float64(-z) * Float64(z / Float64(y - z))))) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-54], N[(x + 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[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] + N[((-z) * N[(z / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.1%

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

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   if -2e-3 < (tan.f64 a) < 1e-54

   1. Initial program 78.5%

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

      1. lift-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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1e-54 < (tan.f64 a)

   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

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 80.2

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

   4. Applied rewrites 80.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;\left(\tan \left(z + y\right) + x\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-54}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{fma}\left(y, \frac{y}{y - z}, \left(-z\right) \cdot \frac{z}{y - z}\right)\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-10}:\\ \;\;\;\;x + {\left({\left(\tan \left(z + y\right) - \tan a\right)}^{-1}\right)}^{-1}\\ \mathbf{elif}\;\tan a \leq 10^{-54}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{fma}\left(y, \frac{y}{y - z}, \left(-z\right) \cdot \frac{z}{y - z}\right)\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -2e-10)
   (+ x (pow (pow (- (tan (+ z y)) (tan a)) -1.0) -1.0))
   (if (<= (tan a) 1e-54)
     (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- x))
     (+ x (- (tan (fma y (/ y (- y z)) (* (- z) (/ z (- y z))))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -2e-10) {
		tmp = x + pow(pow((tan((z + y)) - tan(a)), -1.0), -1.0);
	} else if (tan(a) <= 1e-54) {
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - -x;
	} else {
		tmp = x + (tan(fma(y, (y / (y - z)), (-z * (z / (y - z))))) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -2e-10)
		tmp = Float64(x + ((Float64(tan(Float64(z + y)) - tan(a)) ^ -1.0) ^ -1.0));
	elseif (tan(a) <= 1e-54)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - Float64(-x));
	else
		tmp = Float64(x + Float64(tan(fma(y, Float64(y / Float64(y - z)), Float64(Float64(-z) * Float64(z / Float64(y - z))))) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -2e-10], N[(x + N[Power[N[Power[N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-54], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], N[(x + N[(N[Tan[N[(y * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] + N[((-z) * N[(z / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2.00000000000000007e-10

   1. Initial program 77.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 77.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 77.9

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

   4. Applied rewrites 77.9%

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

   if -2.00000000000000007e-10 < (tan.f64 a) < 1e-54

   1. Initial program 78.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 78.1

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

   7. Applied rewrites 78.1%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(-x\right)} \]
   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. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower--.f64 99.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 1e-54 < (tan.f64 a)

   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

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 80.2

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

   4. Applied rewrites 80.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-10}:\\ \;\;\;\;x + {\left({\left(\tan \left(z + y\right) - \tan a\right)}^{-1}\right)}^{-1}\\ \mathbf{elif}\;\tan a \leq 10^{-54}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{fma}\left(y, \frac{y}{y - z}, \left(-z\right) \cdot \frac{z}{y - z}\right)\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Final simplification 78.7%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

Alternative 6: 50.7% 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 78.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-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 78.6

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
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 51.0

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

7. Applied rewrites 51.0%

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

Reproduce

herbie shell --seed 2024309 
(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))))