tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 38.4s

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: 79.1% 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 77.7%

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

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\_0 \cdot \frac{-1}{-1 + \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -4e-13) (not (<= (tan a) 5e-13)))
     (fma 1.0 t_0 (- x (tan a)))
     (+ x (* t_0 (/ -1.0 (+ -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 ((tan(a) <= -4e-13) || !(tan(a) <= 5e-13)) {
		tmp = fma(1.0, t_0, (x - tan(a)));
	} else {
		tmp = x + (t_0 * (-1.0 / (-1.0 + (tan(y) * tan(z)))));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -4e-13) || !(tan(a) <= 5e-13))
		tmp = fma(1.0, t_0, Float64(x - tan(a)));
	else
		tmp = Float64(x + Float64(t_0 * Float64(-1.0 / Float64(-1.0 + Float64(tan(y) * tan(z))))));
	end
	return 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[N[Tan[a], $MachinePrecision], -4e-13], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-13]], $MachinePrecision]], N[(1.0 * t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 * 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 4.9999999999999999e-13 < (tan.f64 a)

   1. Initial program 75.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-neg 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 76.0%

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

   if -4.0000000000000001e-13 < (tan.f64 a) < 4.9999999999999999e-13

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. associate-+l- 80.1%

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

   4. Applied egg-rr 80.1%

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

   5. Taylor expanded in a around 0 80.1%

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

      1. neg-mul-1 80.1%

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

   7. Simplified 80.1%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 87.4%

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

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -4 \cdot 10^{-13} \lor \neg \left(\tan a \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t\_0}{-1 + \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -4e-13) (not (<= (tan a) 5e-13)))
     (fma 1.0 t_0 (- x (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 ((tan(a) <= -4e-13) || !(tan(a) <= 5e-13)) {
		tmp = fma(1.0, t_0, (x - tan(a)));
	} else {
		tmp = x - (t_0 / (-1.0 + (tan(y) * tan(z))));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -4e-13) || !(tan(a) <= 5e-13))
		tmp = fma(1.0, t_0, Float64(x - tan(a)));
	else
		tmp = Float64(x - Float64(t_0 / Float64(-1.0 + Float64(tan(y) * tan(z)))));
	end
	return 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[N[Tan[a], $MachinePrecision], -4e-13], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-13]], $MachinePrecision]], N[(1.0 * t$95$0 + N[(x - 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 (tan.f64 a) < -4.0000000000000001e-13 or 4.9999999999999999e-13 < (tan.f64 a)

   1. Initial program 75.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-neg 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0 76.0%

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

   if -4.0000000000000001e-13 < (tan.f64 a) < 4.9999999999999999e-13

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. associate-+l- 80.1%

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

   4. Applied egg-rr 80.1%

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

   5. Taylor expanded in a around 0 80.1%

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

      1. neg-mul-1 80.1%

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

   7. Simplified 80.1%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.7%

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

       2. *-rgt-identity 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 87.4%

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

Alternative 4: 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 77.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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/ 60.8%

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

   2. *-rgt-identity 60.8%

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

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%

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

Alternative 5: 79.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

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

   3. *-commutative 99.7%

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

   4. fma-neg 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in y around 0 77.9%

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

8. Final simplification 77.9%

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

Alternative 6: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-9}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-9) (+ x (tan (+ y z))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-9) {
		tmp = x + tan((y + z));
	} else {
		tmp = tan(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) :: tmp
    if ((y + z) <= (-2d-9)) then
        tmp = x + tan((y + z))
    else
        tmp = tan(z) + (x - 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) <= -2e-9) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2e-9:
		tmp = x + math.tan((y + z))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-9)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2e-9)
		tmp = x + tan((y + z));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-9], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -2.00000000000000012e-9

   1. Initial program 74.4%

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

      1. +-commutative 74.4%

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

      2. associate-+l- 74.2%

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

   4. Applied egg-rr 74.2%

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

   5. Taylor expanded in a around 0 47.4%

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

      1. neg-mul-1 47.4%

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

   7. Simplified 47.4%

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

      1. sub-neg 47.4%

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

      2. +-commutative 47.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 3.9%

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

      5. sqr-neg 3.9%

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

      6. sqrt-unprod 3.9%

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

      7. add-sqr-sqrt 3.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 47.4%

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

      10. sqr-neg 47.4%

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

      11. sqrt-unprod 47.2%

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

      12. add-sqr-sqrt 47.4%

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

   9. Applied egg-rr 47.4%

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

   if -2.00000000000000012e-9 < (+.f64 y z)

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. associate--l+ 66.4%

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

   5. Simplified 66.4%

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

      1. tan-quot 66.4%

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

      2. tan-quot 66.4%

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

      3. associate-+r- 66.4%

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

   7. Applied egg-rr 66.4%

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

      1. associate-+r- 66.4%

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

   9. Simplified 66.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-9}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-11) (+ x (- (tan y) (tan a))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = tan(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) :: tmp
    if ((y + z) <= (-5d-11)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = tan(z) + (x - 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) <= -5e-11) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5e-11:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5e-11)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5e-11)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5e-11], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -5.00000000000000018e-11

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. associate-+l- 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in z around 0 48.1%

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

      1. tan-quot 48.2%

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

      2. associate--r- 48.3%

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

   7. Applied egg-rr 48.3%

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

   if -5.00000000000000018e-11 < (+.f64 y z)

   1. Initial program 79.7%

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

   3. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. associate--l+ 66.3%

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

   5. Simplified 66.3%

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

      1. tan-quot 66.3%

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

      2. tan-quot 66.3%

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

      3. associate-+r- 66.3%

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

   7. Applied egg-rr 66.3%

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

      1. associate-+r- 66.3%

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

   9. Simplified 66.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% 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 77.7%

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

3. Final simplification 77.7%

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

Alternative 9: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000 \lor \neg \left(y + z \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -2000.0) (not (<= (+ y z) 4e-12)))
   (+ x (tan (+ y z)))
   (+ y (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -2000.0) || !((y + z) <= 4e-12)) {
		tmp = x + tan((y + z));
	} else {
		tmp = y + (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) :: tmp
    if (((y + z) <= (-2000.0d0)) .or. (.not. ((y + z) <= 4d-12))) then
        tmp = x + tan((y + z))
    else
        tmp = y + (x - 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) <= -2000.0) || !((y + z) <= 4e-12)) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = y + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -2000.0) or not ((y + z) <= 4e-12):
		tmp = x + math.tan((y + z))
	else:
		tmp = y + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -2000.0) || !(Float64(y + z) <= 4e-12))
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(y + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -2000.0) || ~(((y + z) <= 4e-12)))
		tmp = x + tan((y + z));
	else
		tmp = y + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -2000.0], N[Not[LessEqual[N[(y + z), $MachinePrecision], 4e-12]], $MachinePrecision]], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -2e3 or 3.99999999999999992e-12 < (+.f64 y z)

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. associate-+l- 69.8%

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in a around 0 47.1%

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

      1. neg-mul-1 47.1%

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

   7. Simplified 47.1%

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

      1. sub-neg 47.1%

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

      2. +-commutative 47.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 4.1%

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

      5. sqr-neg 4.1%

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

      6. sqrt-unprod 4.1%

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

      7. add-sqr-sqrt 4.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 47.1%

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

      10. sqr-neg 47.1%

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

      11. sqrt-unprod 46.9%

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

      12. add-sqr-sqrt 47.1%

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

   9. Applied egg-rr 47.1%

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

   if -2e3 < (+.f64 y z) < 3.99999999999999992e-12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l- 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0 99.2%

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

   6. Taylor expanded in y around 0 99.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000 \lor \neg \left(y + z \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% 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 77.7%

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

   1. +-commutative 77.7%

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

   2. associate-+l- 77.7%

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

4. Applied egg-rr 77.7%

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

5. Taylor expanded in a around 0 51.3%

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

   1. neg-mul-1 51.3%

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

7. Simplified 51.3%

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

   1. sub-neg 51.3%

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

   2. +-commutative 51.3%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 3.5%

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

   5. sqr-neg 3.5%

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

   6. sqrt-unprod 3.5%

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

   7. add-sqr-sqrt 3.5%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 51.3%

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

   10. sqr-neg 51.3%

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

   11. sqrt-unprod 51.1%

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

   12. add-sqr-sqrt 51.3%

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

9. Applied egg-rr 51.3%

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

10. Final simplification 51.3%

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

Alternative 11: 32.0% 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 77.7%

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

3. Taylor expanded in x around inf 32.9%

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

4. Final simplification 32.9%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))