tan-example (used to crash)

Percentage Accurate: 78.9% → 99.7%

Time: 32.7s

Alternatives: 10

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: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(fma(sin(z), (cos(z) ^ -1.0), 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[Sin[z], $MachinePrecision] * N[Power[N[Cos[z], $MachinePrecision], -1.0], $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 76.9%

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

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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)) - (Float64(cos(a) / sin(a)) ^ -1.0)))
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[Power[N[(N[Cos[a], $MachinePrecision] / N[Sin[a], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

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

4. Applied rewrites 99.6%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-sin.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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) ^ -1.0) ^ -1.0)))
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[Power[N[Power[N[Tan[a], $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

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

4. Applied rewrites 99.6%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-sin.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-tan.f64 N/A

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

   7. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_0 - \tan y, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(t\_0, \tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))))
   (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-5)))
     (fma (- t_0 (tan y)) (pow -1.0 -1.0) (- x (tan a)))
     (+
      x
      (-
       (/ (+ (tan z) (tan y)) (fma t_0 (tan y) 1.0))
       (* (fma (* a a) 0.3333333333333333 1.0) a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-5)) {
		tmp = fma((t_0 - tan(y)), pow(-1.0, -1.0), (x - tan(a)));
	} else {
		tmp = x + (((tan(z) + tan(y)) / fma(t_0, tan(y), 1.0)) - (fma((a * a), 0.3333333333333333, 1.0) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-5))
		tmp = fma(Float64(t_0 - tan(y)), (-1.0 ^ -1.0), Float64(x - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / fma(t_0, tan(y), 1.0)) - Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-5]], $MachinePrecision]], N[(N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[Power[-1.0, -1.0], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 1.00000000000000008e-5 < (tan.f64 a)

   1. Initial program 79.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 79.8%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1.00000000000000008e-5

   1. Initial program 74.8%

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

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

   7. Applied rewrites 99.7%

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

4. Final simplification 90.4%

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

Alternative 5: 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 76.9%

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

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

4. Applied rewrites 99.6%

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

Alternative 6: 79.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-+.f64 N/A

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

   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 y around 0

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

7. Applied rewrites 77.3%

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

8. Final simplification 77.3%

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

Alternative 7: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550:\\ \;\;\;\;\tan \left(\mathsf{fma}\left(\frac{z}{y}, y, y\right)\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -550.0)
   (- (tan (fma (/ z y) y y)) (- x))
   (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -550.0) {
		tmp = tan(fma((z / y), y, y)) - -x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -550.0)
		tmp = Float64(tan(fma(Float64(z / y), y, y)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -550.0], N[(N[Tan[N[(N[(z / y), $MachinePrecision] * y + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -550

   1. Initial program 59.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-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 59.8

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

   4. Applied rewrites 59.8%

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

   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 47.8

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

   7. Applied rewrites 47.8%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 47.8

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

   10. Applied rewrites 47.8%

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

   if -550 < y

   1. Initial program 84.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 72.1

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

   7. Applied rewrites 72.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 72.1

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

   9. Applied rewrites 72.1%

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

Alternative 8: 78.9% 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 76.9%

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

Alternative 9: 31.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left({x}^{-1}\right)}^{-1} \end{array} \]

FPCore

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

C

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

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)) ** (-1.0d0)
end function

Java

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

Python

def code(x, y, z, a):
	return math.pow(math.pow(x, -1.0), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 76.7

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 29.6

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

7. Applied rewrites 29.6%

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

8. Final simplification 29.6%

   \[\leadsto {\left({x}^{-1}\right)}^{-1} \]
9. Add Preprocessing

Alternative 10: 50.3% 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 76.9%

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

   1. lift-+.f64 N/A

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

   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 76.8

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

4. Applied rewrites 76.8%

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

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 50.0

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

7. Applied rewrites 50.0%

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

Reproduce

herbie shell --seed 2024320 
(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))))