tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 28.0s

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: 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

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- (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)) / -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(-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[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

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

3. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
4. 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%

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 93.8%

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

   2. expm1-udef 93.8%

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

   3. log1p-udef 93.8%

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

   4. add-exp-log 99.6%

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

7. Applied egg-rr 99.6%

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

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

9. Simplified 99.7%

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

    1. sub-neg 99.7%

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

    2. associate--r+ 99.7%

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

    3. metadata-eval 99.7%

       \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{0} - \mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(-\tan a\right)\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto x + \color{blue}{\left(\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(-\tan a\right)\right)} \]
12. Step-by-step derivation

    1. sub-neg 99.7%

       \[\leadsto x + \color{blue}{\left(\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)} - \tan a\right)} \]

    2. sub0-neg 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 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 80.3%

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

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

3. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
4. 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%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-9} \lor \neg \left(a \leq 2.9 \cdot 10^{-31}\right):\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= a -6.5e-9) (not (<= a 2.9e-31)))
     (+ x (- t_0 (tan a)))
     (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((a <= -6.5e-9) || !(a <= 2.9e-31)) {
		tmp = x + (t_0 - tan(a));
	} else {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if ((a <= (-6.5d-9)) .or. (.not. (a <= 2.9d-31))) then
        tmp = x + (t_0 - tan(a))
    else
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if ((a <= -6.5e-9) || !(a <= 2.9e-31)) {
		tmp = x + (t_0 - Math.tan(a));
	} else {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if (a <= -6.5e-9) or not (a <= 2.9e-31):
		tmp = x + (t_0 - math.tan(a))
	else:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((a <= -6.5e-9) || !(a <= 2.9e-31))
		tmp = Float64(x + Float64(t_0 - tan(a)));
	else
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if ((a <= -6.5e-9) || ~((a <= 2.9e-31)))
		tmp = x + (t_0 - tan(a));
	else
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -6.5e-9], N[Not[LessEqual[a, 2.9e-31]], $MachinePrecision]], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000003e-9 or 2.9000000000000001e-31 < a

   1. Initial program 78.3%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 78.8%

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

   if -6.5000000000000003e-9 < a < 2.9000000000000001e-31

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. associate-+l- 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in a around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   6. Simplified 82.4%

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

      1. sub-neg 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. remove-double-neg 82.4%

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

      2. +-commutative 82.4%

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

      3. +-commutative 82.4%

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

   10. Simplified 82.4%

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

       1. tan-sum 99.7%

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

       2. +-commutative 99.7%

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

   12. Applied egg-rr 99.7%

       \[\leadsto x + \color{blue}{\frac{\tan y + \tan z}{1 - \tan z \cdot \tan y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-9} \lor \neg \left(a \leq 2.9 \cdot 10^{-31}\right):\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]

Alternative 4: 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 80.3%

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

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

3. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
4. 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%

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

5. Simplified 99.7%

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

6. Taylor expanded in y around 0 80.7%

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

7. Final simplification 80.7%

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

Alternative 5: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-41) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-41) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	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 ((y + z) <= 4d-41) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-41) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 4e-41:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 4e-41)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 4e-41)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 4e-41], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 4.00000000000000002e-41

   1. Initial program 83.5%

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

   2. Taylor expanded in z around 0 66.1%

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

      1. tan-quot 66.1%

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

      2. *-un-lft-identity 66.1%

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

   4. Applied egg-rr 66.1%

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

      1. *-lft-identity 66.1%

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

   6. Simplified 66.1%

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

   if 4.00000000000000002e-41 < (+.f64 y z)

   1. Initial program 73.8%

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

      1. +-commutative 73.8%

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

      2. associate-+l- 73.7%

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

   3. Applied egg-rr 73.7%

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

   4. Taylor expanded in a around 0 47.7%

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

      1. neg-mul-1 47.7%

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

   6. Simplified 47.7%

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

      1. sub-neg 47.7%

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

   8. Applied egg-rr 47.7%

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

      1. remove-double-neg 47.7%

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

      2. +-commutative 47.7%

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

      3. +-commutative 47.7%

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

   10. Simplified 47.7%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 6: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin z}{\cos z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-41) (+ x (- (tan y) (tan a))) (+ x (/ (sin z) (cos z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-41) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (sin(z) / cos(z));
	}
	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 ((y + z) <= 4d-41) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (sin(z) / cos(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-41) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.sin(z) / Math.cos(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 4e-41:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.sin(z) / math.cos(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 4e-41)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(sin(z) / cos(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 4e-41)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (sin(z) / cos(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 4e-41], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 4.00000000000000002e-41

   1. Initial program 83.5%

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

   2. Taylor expanded in z around 0 66.1%

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

      1. tan-quot 66.1%

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

      2. *-un-lft-identity 66.1%

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

   4. Applied egg-rr 66.1%

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

      1. *-lft-identity 66.1%

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

   6. Simplified 66.1%

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

   if 4.00000000000000002e-41 < (+.f64 y z)

   1. Initial program 73.8%

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

      1. +-commutative 73.8%

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

      2. associate-+l- 73.7%

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

   3. Applied egg-rr 73.7%

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

   4. Taylor expanded in a around 0 47.7%

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

      1. neg-mul-1 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in y around 0 32.7%

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin z}{\cos z}\\ \end{array} \]

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 80.3%

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

2. Final simplification 80.3%

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

Alternative 8: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -200000000 \lor \neg \left(y + z \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -200000000.0) (not (<= (+ y z) 4e-41)))
   (+ x (tan (+ y z)))
   (+ x (- y (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -200000000.0) || !((y + z) <= 4e-41)) {
		tmp = x + tan((y + z));
	} else {
		tmp = x + (y - tan(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 (((y + z) <= (-200000000.0d0)) .or. (.not. ((y + z) <= 4d-41))) then
        tmp = x + tan((y + z))
    else
        tmp = x + (y - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -200000000.0) || !((y + z) <= 4e-41)) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + (y - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -200000000.0) or not ((y + z) <= 4e-41):
		tmp = x + math.tan((y + z))
	else:
		tmp = x + (y - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -200000000.0) || !(Float64(y + z) <= 4e-41))
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(x + Float64(y - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -200000000.0) || ~(((y + z) <= 4e-41)))
		tmp = x + tan((y + z));
	else
		tmp = x + (y - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -200000000.0], N[Not[LessEqual[N[(y + z), $MachinePrecision], 4e-41]], $MachinePrecision]], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -2e8 or 4.00000000000000002e-41 < (+.f64 y z)

   1. Initial program 74.2%

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

      1. +-commutative 74.2%

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

      2. associate-+l- 74.1%

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

   3. Applied egg-rr 74.1%

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

   4. Taylor expanded in a around 0 48.4%

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

      1. neg-mul-1 48.4%

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

   6. Simplified 48.4%

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

      1. sub-neg 48.4%

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

   8. Applied egg-rr 48.4%

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

      1. remove-double-neg 48.4%

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

      2. +-commutative 48.4%

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

      3. +-commutative 48.4%

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

   10. Simplified 48.4%

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

   if -2e8 < (+.f64 y z) < 4.00000000000000002e-41

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.4%

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

   3. Taylor expanded in y around 0 98.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -200000000 \lor \neg \left(y + z \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]

Alternative 9: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

   2. associate-+l- 80.2%

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

3. Applied egg-rr 80.2%

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

4. Taylor expanded in a around 0 53.8%

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

   1. neg-mul-1 53.8%

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

6. Simplified 53.8%

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

   1. sub-neg 53.8%

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

8. Applied egg-rr 53.8%

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

   1. remove-double-neg 53.8%

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

   2. +-commutative 53.8%

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

   3. +-commutative 53.8%

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

10. Simplified 53.8%

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

11. Final simplification 53.8%

    \[\leadsto x + \tan \left(y + z\right) \]

Alternative 10: 31.6% 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 80.3%

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

2. Taylor expanded in x around inf 33.8%

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

3. Final simplification 33.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023195 
(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))))