tan-example (used to crash)

Percentage Accurate: 80.0% → 99.7%

Time: 44.0s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.0%

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

   1. tan-sum 99.5%

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

   2. div-inv 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan y + \tan z\right) \cdot \frac{-1}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -4e-13) (not (<= (tan a) 4e-29)))
   (+ x (- (tan (+ y z)) (tan a)))
   (- x (* (+ (tan y) (tan z)) (/ -1.0 (- 1.0 (* (tan y) (tan z))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -4e-13) || !(tan(a) <= 4e-29)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = x - ((tan(y) + tan(z)) * (-1.0 / (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) :: tmp
    if ((tan(a) <= (-4d-13)) .or. (.not. (tan(a) <= 4d-29))) then
        tmp = x + (tan((y + z)) - tan(a))
    else
        tmp = x - ((tan(y) + tan(z)) * ((-1.0d0) / (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 tmp;
	if ((Math.tan(a) <= -4e-13) || !(Math.tan(a) <= 4e-29)) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else {
		tmp = x - ((Math.tan(y) + Math.tan(z)) * (-1.0 / (1.0 - (Math.tan(y) * Math.tan(z)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -4e-13) or not (math.tan(a) <= 4e-29):
		tmp = x + (math.tan((y + z)) - math.tan(a))
	else:
		tmp = x - ((math.tan(y) + math.tan(z)) * (-1.0 / (1.0 - (math.tan(y) * math.tan(z)))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -4e-13) || !(tan(a) <= 4e-29))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(x - Float64(Float64(tan(y) + tan(z)) * Float64(-1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -4e-13) || ~((tan(a) <= 4e-29)))
		tmp = x + (tan((y + z)) - tan(a));
	else
		tmp = x - ((tan(y) + tan(z)) * (-1.0 / (1.0 - (tan(y) * tan(z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -4e-13], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 4e-29]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.0000000000000001e-13 or 3.99999999999999977e-29 < (tan.f64 a)

   1. Initial program 79.9%

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

   if -4.0000000000000001e-13 < (tan.f64 a) < 3.99999999999999977e-29

   1. Initial program 71.6%

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

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

      2. un-div-inv 99.7%

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

      3. tan-sum 71.6%

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

      4. associate--r- 71.6%

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

      5. tan-sum 99.7%

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

      6. un-div-inv 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-neg 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. neg-mul-1 71.6%

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

   9. Simplified 99.7%

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

       1. fma-undefine 99.7%

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

       2. remove-double-neg 99.7%

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

   11. Applied egg-rr 99.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan y + \tan z\right) \cdot \frac{-1}{1 - \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -4e-13) (not (<= (tan a) 4e-29)))
   (+ x (- (tan (+ y z)) (tan a)))
   (- x (/ (+ (tan y) (tan z)) (+ -1.0 (* (tan y) (tan z)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -4e-13) || !(tan(a) <= 4e-29)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = x - ((tan(y) + tan(z)) / (-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) :: tmp
    if ((tan(a) <= (-4d-13)) .or. (.not. (tan(a) <= 4d-29))) then
        tmp = x + (tan((y + z)) - tan(a))
    else
        tmp = x - ((tan(y) + tan(z)) / ((-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 tmp;
	if ((Math.tan(a) <= -4e-13) || !(Math.tan(a) <= 4e-29)) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else {
		tmp = x - ((Math.tan(y) + Math.tan(z)) / (-1.0 + (Math.tan(y) * Math.tan(z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -4e-13) or not (math.tan(a) <= 4e-29):
		tmp = x + (math.tan((y + z)) - math.tan(a))
	else:
		tmp = x - ((math.tan(y) + math.tan(z)) / (-1.0 + (math.tan(y) * math.tan(z))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -4e-13) || !(tan(a) <= 4e-29))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(x - Float64(Float64(tan(y) + tan(z)) / Float64(-1.0 + Float64(tan(y) * tan(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -4e-13) || ~((tan(a) <= 4e-29)))
		tmp = x + (tan((y + z)) - tan(a));
	else
		tmp = x - ((tan(y) + tan(z)) / (-1.0 + (tan(y) * tan(z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -4e-13], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 4e-29]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -4.0000000000000001e-13 or 3.99999999999999977e-29 < (tan.f64 a)

   1. Initial program 79.9%

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

   if -4.0000000000000001e-13 < (tan.f64 a) < 3.99999999999999977e-29

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. associate-+l- 71.6%

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

   4. Applied egg-rr 71.6%

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

   5. Taylor expanded in a around 0 71.6%

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

      1. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.7%

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

      4. remove-double-neg 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. fma-undefine 99.7%

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

       2. *-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. associate-*l/ 99.7%

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

       5. *-lft-identity 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-13} \lor \neg \left(a \leq 5 \cdot 10^{-27}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{-1}{-1 + \tan y \cdot \tan z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -4e-13) (not (<= a 5e-27)))
   (+ x (- (tan (+ y z)) (tan a)))
   (fma (+ (tan y) (tan z)) (/ -1.0 (+ -1.0 (* (tan y) (tan z)))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -4e-13) || !(a <= 5e-27)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = fma((tan(y) + tan(z)), (-1.0 / (-1.0 + (tan(y) * tan(z)))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -4e-13) || !(a <= 5e-27))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = fma(Float64(tan(y) + tan(z)), Float64(-1.0 / Float64(-1.0 + Float64(tan(y) * tan(z)))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -4e-13], N[Not[LessEqual[a, 5e-27]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.0000000000000001e-13 or 5.0000000000000002e-27 < a

   1. Initial program 79.9%

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

   if -4.0000000000000001e-13 < a < 5.0000000000000002e-27

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. associate-+l- 71.6%

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

   4. Applied egg-rr 71.6%

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

   5. Taylor expanded in a around 0 71.6%

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

      1. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.7%

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

      4. remove-double-neg 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-13} \lor \neg \left(a \leq 5 \cdot 10^{-27}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{-1}{-1 + \tan y \cdot \tan z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - (tan(a) + ((tan(y) + tan(z)) / (-1.0 + (tan(y) * tan(z)))));
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[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. tan-sum 99.5%

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

   2. div-inv 99.5%

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

4. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + t\_0\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x t_0)))
   (if (<= t_0 -5e-7) t_1 (if (<= t_0 2e-25) (+ z (- x (tan a))) (fabs t_1)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_1;
	} else if (t_0 <= 2e-25) {
		tmp = z + (x - tan(a));
	} else {
		tmp = fabs(t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = tan((y + z))
    t_1 = x + t_0
    if (t_0 <= (-5d-7)) then
        tmp = t_1
    else if (t_0 <= 2d-25) then
        tmp = z + (x - tan(a))
    else
        tmp = abs(t_1)
    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 + z));
	double t_1 = x + t_0;
	double tmp;
	if (t_0 <= -5e-7) {
		tmp = t_1;
	} else if (t_0 <= 2e-25) {
		tmp = z + (x - Math.tan(a));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	t_1 = x + t_0
	tmp = 0
	if t_0 <= -5e-7:
		tmp = t_1
	elif t_0 <= 2e-25:
		tmp = z + (x - math.tan(a))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (t_0 <= -5e-7)
		tmp = t_1;
	elseif (t_0 <= 2e-25)
		tmp = Float64(z + Float64(x - tan(a)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	t_1 = x + t_0;
	tmp = 0.0;
	if (t_0 <= -5e-7)
		tmp = t_1;
	elseif (t_0 <= 2e-25)
		tmp = z + (x - tan(a));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-7], t$95$1, If[LessEqual[t$95$0, 2e-25], N[(z + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 (+.f64 y z)) < -4.99999999999999977e-7

   1. Initial program 69.2%

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

      1. +-commutative 69.2%

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

      2. associate-+l- 69.1%

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

   4. Applied egg-rr 69.1%

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

   5. Taylor expanded in a around 0 41.6%

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

      1. neg-mul-1 41.6%

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

   7. Simplified 41.6%

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

      1. sub-neg 41.6%

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

      2. remove-double-neg 41.6%

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

   9. Applied egg-rr 41.6%

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

   if -4.99999999999999977e-7 < (tan.f64 (+.f64 y z)) < 2.00000000000000008e-25

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+l- 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0 99.4%

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

   6. Taylor expanded in z around 0 99.4%

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

   if 2.00000000000000008e-25 < (tan.f64 (+.f64 y z))

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. associate-+l- 71.6%

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

   4. Applied egg-rr 71.6%

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

   5. Taylor expanded in a around 0 46.3%

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

      1. neg-mul-1 46.3%

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

   7. Simplified 46.3%

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

      1. add-sqr-sqrt 46.1%

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

      2. sqrt-unprod 46.3%

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

      3. pow2 46.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\tan \left(y + z\right) - \left(-x\right)\right)}^{2}}} \]

      4. sub-neg 46.3%

         \[\leadsto \sqrt{{\color{blue}{\left(\tan \left(y + z\right) + \left(-\left(-x\right)\right)\right)}}^{2}} \]

      5. remove-double-neg 46.3%

         \[\leadsto \sqrt{{\left(\tan \left(y + z\right) + \color{blue}{x}\right)}^{2}} \]

   9. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\tan \left(y + z\right) + x\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 46.3%

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

       2. rem-sqrt-square 46.3%

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

       3. +-commutative 46.3%

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

   11. Simplified 46.3%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -5 \cdot 10^{-7}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;\tan \left(y + z\right) \leq 2 \cdot 10^{-25}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x + \tan \left(y + z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-7} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-25}\right):\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (or (<= t_0 -5e-7) (not (<= t_0 2e-25)))
     (+ x t_0)
     (+ z (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if ((t_0 <= -5e-7) || !(t_0 <= 2e-25)) {
		tmp = x + t_0;
	} else {
		tmp = z + (x - 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) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if ((t_0 <= (-5d-7)) .or. (.not. (t_0 <= 2d-25))) then
        tmp = x + t_0
    else
        tmp = z + (x - tan(a))
    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 + z));
	double tmp;
	if ((t_0 <= -5e-7) || !(t_0 <= 2e-25)) {
		tmp = x + t_0;
	} else {
		tmp = z + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if (t_0 <= -5e-7) or not (t_0 <= 2e-25):
		tmp = x + t_0
	else:
		tmp = z + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if ((t_0 <= -5e-7) || !(t_0 <= 2e-25))
		tmp = Float64(x + t_0);
	else
		tmp = Float64(z + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if ((t_0 <= -5e-7) || ~((t_0 <= 2e-25)))
		tmp = x + t_0;
	else
		tmp = z + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-7], N[Not[LessEqual[t$95$0, 2e-25]], $MachinePrecision]], N[(x + t$95$0), $MachinePrecision], N[(z + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -4.99999999999999977e-7 or 2.00000000000000008e-25 < (tan.f64 (+.f64 y z))

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-+l- 70.3%

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in a around 0 43.8%

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

      1. neg-mul-1 43.8%

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

   7. Simplified 43.8%

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

      1. sub-neg 43.8%

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

      2. remove-double-neg 43.8%

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

   9. Applied egg-rr 43.8%

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

   if -4.99999999999999977e-7 < (tan.f64 (+.f64 y z)) < 2.00000000000000008e-25

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+l- 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0 99.4%

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

   6. Taylor expanded in z around 0 99.4%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -5 \cdot 10^{-7} \lor \neg \left(\tan \left(y + z\right) \leq 2 \cdot 10^{-25}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -7.2e-7) (+ x (/ (sin y) (cos y))) (- x (- (tan a) (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -7.2e-7) {
		tmp = x + (sin(y) / cos(y));
	} else {
		tmp = x - (tan(a) - 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) :: tmp
    if (y <= (-7.2d-7)) then
        tmp = x + (sin(y) / cos(y))
    else
        tmp = x - (tan(a) - tan(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -7.2e-7) {
		tmp = x + (Math.sin(y) / Math.cos(y));
	} else {
		tmp = x - (Math.tan(a) - Math.tan(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -7.2e-7:
		tmp = x + (math.sin(y) / math.cos(y))
	else:
		tmp = x - (math.tan(a) - math.tan(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -7.2e-7)
		tmp = Float64(x + Float64(sin(y) / cos(y)));
	else
		tmp = Float64(x - Float64(tan(a) - tan(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -7.2e-7)
		tmp = x + (sin(y) / cos(y));
	else
		tmp = x - (tan(a) - tan(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -7.2e-7], N[(x + N[(N[Sin[y], $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Tan[a], $MachinePrecision] - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999989e-7

   1. Initial program 57.9%

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

      1. +-commutative 57.9%

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

      2. associate-+l- 57.9%

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

   4. Applied egg-rr 57.9%

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

   5. Taylor expanded in a around 0 35.7%

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

      1. neg-mul-1 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in z around 0 35.5%

      \[\leadsto \color{blue}{x + \frac{\sin y}{\cos y}} \]

   if -7.19999999999999989e-7 < y

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. associate-+l- 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in y around 0 69.2%

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

      1. tan-quot 69.3%

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

      2. associate--r- 69.3%

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

   7. Applied egg-rr 69.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{\sin y}{\cos y}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Final simplification 76.0%

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

Alternative 10: 51.1% 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 76.0%

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

   1. +-commutative 76.0%

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

   2. associate-+l- 76.0%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in a around 0 46.5%

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

   1. neg-mul-1 46.5%

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

7. Simplified 46.5%

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

   1. sub-neg 46.5%

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

   2. remove-double-neg 46.5%

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

9. Applied egg-rr 46.5%

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

10. Final simplification 46.5%

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

Alternative 11: 31.9% 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 76.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 2024055 
(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))))