tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 38.6s

Alternatives: 13

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: 79.5% 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

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\tan y + \tan z, \frac{1}{\left(1 - \left(2 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right) - -1}, -\tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma
   (+ (tan y) (tan z))
   (/ 1.0 (- (- 1.0 (+ 2.0 (fma (tan y) (tan z) -1.0))) -1.0))
   (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma((tan(y) + tan(z)), (1.0 / ((1.0 - (2.0 + fma(tan(y), tan(z), -1.0))) - -1.0)), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(Float64(tan(y) + tan(z)), Float64(1.0 / Float64(Float64(1.0 - Float64(2.0 + fma(tan(y), tan(z), -1.0))) - -1.0)), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(1.0 - N[(2.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-undefine 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan y \cdot \tan z\right)} - 1\right)}} - \tan a\right) \]

   3. log1p-undefine 91.5%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. fma-neg 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

    1. expm1-log1p-u 91.5%

       \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right)}} - \tan a\right) \]

    2. expm1-undefine 91.5%

       \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} - 1\right)}} - \tan a\right) \]

    3. log1p-undefine 91.5%

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

    4. add-exp-log 99.7%

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

12. Applied egg-rr 99.7%

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

    1. sub-neg 99.7%

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

    2. associate-+r+ 99.7%

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

    3. metadata-eval 99.7%

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

    4. metadata-eval 99.7%

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

14. Simplified 99.7%

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

    1. div-inv 99.7%

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

    2. fma-neg 99.7%

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

    3. associate--r+ 99.7%

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

16. Applied egg-rr 99.7%

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

17. Final simplification 99.7%

    \[\leadsto x + \mathsf{fma}\left(\tan y + \tan z, \frac{1}{\left(1 - \left(2 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right) - -1}, -\tan a\right) \]
18. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \left(-1 + \left(2 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right)} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (tan y) (tan z)) (- 1.0 (+ -1.0 (+ 2.0 (fma (tan y) (tan z) -1.0)))))
   (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (-1.0 + (2.0 + fma(tan(y), tan(z), -1.0))))) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(-1.0 + Float64(2.0 + fma(tan(y), tan(z), -1.0))))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-1.0 + N[(2.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-undefine 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan y \cdot \tan z\right)} - 1\right)}} - \tan a\right) \]

   3. log1p-undefine 91.5%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. fma-neg 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

    1. expm1-log1p-u 91.5%

       \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right)}} - \tan a\right) \]

    2. expm1-undefine 91.5%

       \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} - 1\right)}} - \tan a\right) \]

    3. log1p-undefine 91.5%

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

    4. add-exp-log 99.7%

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

12. Applied egg-rr 99.7%

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

    1. sub-neg 99.7%

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

    2. associate-+r+ 99.7%

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

    3. metadata-eval 99.7%

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

    4. metadata-eval 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

    \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \left(-1 + \left(2 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)\right)} - \tan a\right) \]
16. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-undefine 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan y \cdot \tan z\right)} - 1\right)}} - \tan a\right) \]

   3. log1p-undefine 91.5%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. fma-neg 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 79.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. tan-sum 99.7%

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

   2. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-undefine 91.5%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan y \cdot \tan z\right)} - 1\right)}} - \tan a\right) \]

   3. log1p-undefine 91.5%

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

   4. add-exp-log 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. fma-neg 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in y around 0 77.9%

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

12. Final simplification 77.9%

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

Alternative 6: 42.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-42}:\\ \;\;\;\;{e}^{\log x}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -2.4e-42)
   (pow E (log x))
   (if (<= z 8.4e-8) (- (+ x z) (tan a)) (+ x (- (tan z) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -2.4e-42) {
		tmp = pow(((double) M_E), log(x));
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - tan(a);
	} else {
		tmp = x + (tan(z) - a);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -2.4e-42) {
		tmp = Math.pow(Math.E, Math.log(x));
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - Math.tan(a);
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -2.4e-42:
		tmp = math.pow(math.e, math.log(x))
	elif z <= 8.4e-8:
		tmp = (x + z) - math.tan(a)
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -2.4e-42)
		tmp = exp(1) ^ log(x);
	elseif (z <= 8.4e-8)
		tmp = Float64(Float64(x + z) - tan(a));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -2.4e-42)
		tmp = 2.71828182845904523536 ^ log(x);
	elseif (z <= 8.4e-8)
		tmp = (x + z) - tan(a);
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -2.4e-42], N[Power[E, N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 8.4e-8], N[(N[(x + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.40000000000000003e-42

   1. Initial program 52.8%

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

      1. add-exp-log 51.0%

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

      2. +-commutative 51.0%

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

      3. associate-+l- 51.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around inf 22.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.5%

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

      2. log-rec 22.5%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 22.5%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 22.5%

      \[\leadsto e^{\color{blue}{\log x}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 22.5%

         \[\leadsto e^{\color{blue}{1 \cdot \log x}} \]

      2. exp-prod 22.5%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\log x}} \]

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\log x}} \]
   10. Step-by-step derivation

       1. exp-1-e 22.5%

          \[\leadsto {\color{blue}{e}}^{\log x} \]

   11. Simplified 22.5%

       \[\leadsto \color{blue}{{e}^{\log x}} \]

   if -2.40000000000000003e-42 < z < 8.39999999999999978e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.1%

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

   4. Taylor expanded in z around 0 58.1%

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

      1. associate-+r- 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 8.39999999999999978e-8 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0 65.7%

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

   4. Taylor expanded in a around 0 38.9%

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

      1. associate-+r+ 38.9%

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

      2. mul-1-neg 38.9%

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

      3. unsub-neg 38.9%

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

   6. Simplified 38.9%

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

      1. tan-quot 39.0%

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

      2. associate-+l- 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-42}:\\ \;\;\;\;{e}^{\log x}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% 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 77.8%

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

3. Final simplification 77.8%

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

Alternative 8: 42.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-42}:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -2.4e-42)
   (exp (log x))
   (if (<= z 8.4e-8) (- (+ x z) (tan a)) (+ x (- (tan z) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -2.4e-42) {
		tmp = exp(log(x));
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - tan(a);
	} else {
		tmp = x + (tan(z) - a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-2.4d-42)) then
        tmp = exp(log(x))
    else if (z <= 8.4d-8) then
        tmp = (x + z) - tan(a)
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -2.4e-42) {
		tmp = Math.exp(Math.log(x));
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - Math.tan(a);
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -2.4e-42:
		tmp = math.exp(math.log(x))
	elif z <= 8.4e-8:
		tmp = (x + z) - math.tan(a)
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -2.4e-42)
		tmp = exp(log(x));
	elseif (z <= 8.4e-8)
		tmp = Float64(Float64(x + z) - tan(a));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -2.4e-42)
		tmp = exp(log(x));
	elseif (z <= 8.4e-8)
		tmp = (x + z) - tan(a);
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -2.4e-42], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 8.4e-8], N[(N[(x + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.40000000000000003e-42

   1. Initial program 52.8%

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

      1. add-exp-log 51.0%

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

      2. +-commutative 51.0%

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

      3. associate-+l- 51.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around inf 22.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.5%

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

      2. log-rec 22.5%

         \[\leadsto e^{-\color{blue}{\left(-\log x\right)}} \]

      3. remove-double-neg 22.5%

         \[\leadsto e^{\color{blue}{\log x}} \]

   7. Simplified 22.5%

      \[\leadsto e^{\color{blue}{\log x}} \]

   if -2.40000000000000003e-42 < z < 8.39999999999999978e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.1%

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

   4. Taylor expanded in z around 0 58.1%

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

      1. associate-+r- 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 8.39999999999999978e-8 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0 65.7%

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

   4. Taylor expanded in a around 0 38.9%

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

      1. associate-+r+ 38.9%

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

      2. mul-1-neg 38.9%

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

      3. unsub-neg 38.9%

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

   6. Simplified 38.9%

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

      1. tan-quot 39.0%

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

      2. associate-+l- 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-42}:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in y around 0 57.0%

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

   1. tan-quot 57.0%

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

   2. *-un-lft-identity 57.0%

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

5. Applied egg-rr 57.0%

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

   1. *-lft-identity 57.0%

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

7. Simplified 57.0%

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

8. Final simplification 57.0%

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

Alternative 10: 40.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -6.2e-46) x (if (<= z 1.3) (+ x (- z (tan a))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 1.3) {
		tmp = x + (z - tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-6.2d-46)) then
        tmp = x
    else if (z <= 1.3d0) then
        tmp = x + (z - tan(a))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 1.3) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -6.2e-46:
		tmp = x
	elif z <= 1.3:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 1.3)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 1.3)
		tmp = x + (z - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -6.2e-46], x, If[LessEqual[z, 1.3], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000002e-46 or 1.30000000000000004 < z

   1. Initial program 59.0%

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

   3. Taylor expanded in x around inf 21.8%

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

   if -6.2000000000000002e-46 < z < 1.30000000000000004

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0 58.2%

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

   4. Taylor expanded in z around 0 58.2%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -6.2e-46)
   x
   (if (<= z 8.4e-8) (+ x (- z (tan a))) (+ x (- (tan z) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 8.4e-8) {
		tmp = x + (z - tan(a));
	} else {
		tmp = x + (tan(z) - a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-6.2d-46)) then
        tmp = x
    else if (z <= 8.4d-8) then
        tmp = x + (z - tan(a))
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 8.4e-8) {
		tmp = x + (z - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -6.2e-46:
		tmp = x
	elif z <= 8.4e-8:
		tmp = x + (z - math.tan(a))
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 8.4e-8)
		tmp = Float64(x + Float64(z - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 8.4e-8)
		tmp = x + (z - tan(a));
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -6.2e-46], x, If[LessEqual[z, 8.4e-8], N[(x + N[(z - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2000000000000002e-46

   1. Initial program 52.8%

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

   3. Taylor expanded in x around inf 22.5%

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

   if -6.2000000000000002e-46 < z < 8.39999999999999978e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.1%

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

   4. Taylor expanded in z around 0 58.1%

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

   if 8.39999999999999978e-8 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0 65.7%

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

   4. Taylor expanded in a around 0 38.9%

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

      1. associate-+r+ 38.9%

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

      2. mul-1-neg 38.9%

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

      3. unsub-neg 38.9%

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

   6. Simplified 38.9%

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

      1. tan-quot 39.0%

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

      2. associate-+l- 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z -6.2e-46)
   x
   (if (<= z 8.4e-8) (- (+ x z) (tan a)) (+ x (- (tan z) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - tan(a);
	} else {
		tmp = x + (tan(z) - a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-6.2d-46)) then
        tmp = x
    else if (z <= 8.4d-8) then
        tmp = (x + z) - tan(a)
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= -6.2e-46) {
		tmp = x;
	} else if (z <= 8.4e-8) {
		tmp = (x + z) - Math.tan(a);
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= -6.2e-46:
		tmp = x
	elif z <= 8.4e-8:
		tmp = (x + z) - math.tan(a)
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 8.4e-8)
		tmp = Float64(Float64(x + z) - tan(a));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= -6.2e-46)
		tmp = x;
	elseif (z <= 8.4e-8)
		tmp = (x + z) - tan(a);
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, -6.2e-46], x, If[LessEqual[z, 8.4e-8], N[(N[(x + z), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2000000000000002e-46

   1. Initial program 52.8%

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

   3. Taylor expanded in x around inf 22.5%

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

   if -6.2000000000000002e-46 < z < 8.39999999999999978e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.1%

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

   4. Taylor expanded in z around 0 58.1%

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

      1. associate-+r- 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 8.39999999999999978e-8 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0 65.7%

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

   4. Taylor expanded in a around 0 38.9%

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

      1. associate-+r+ 38.9%

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

      2. mul-1-neg 38.9%

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

      3. unsub-neg 38.9%

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

   6. Simplified 38.9%

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

      1. tan-quot 39.0%

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

      2. associate-+l- 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;\left(x + z\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.4% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in x around inf 29.5%

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

4. Final simplification 29.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))