tan-example (used to crash)

Percentage Accurate: 79.5% → 99.6%

Time: 46.6s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\tan y + \tan z, \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{-1 + \tan y \cdot \tan z}\right)\right), -\tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma
   (+ (tan y) (tan z))
   (log1p (expm1 (/ -1.0 (+ -1.0 (* (tan y) (tan z))))))
   (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma((tan(y) + tan(z)), log1p(expm1((-1.0 / (-1.0 + (tan(y) * tan(z)))))), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(Float64(tan(y) + tan(z)), log1p(expm1(Float64(-1.0 / Float64(-1.0 + Float64(tan(y) * tan(z)))))), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[(-1.0 / N[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

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

   3. fmm-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. log1p-expm1-u 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto x + \mathsf{fma}\left(\tan y + \tan z, \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{-1 + \tan y \cdot \tan z}\right)\right), -\tan a\right) \]
8. Add Preprocessing

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 5 \cdot 10^{-24}\right):\\ \;\;\;\;x + \mathsf{fma}\left(t\_0, 1, -\tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{t\_0}{-1 + \tan y \cdot \tan z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -0.005) (not (<= (tan a) 5e-24)))
     (+ x (fma t_0 1.0 (- (tan a))))
     (- x (+ a (/ 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) <= -0.005) || !(tan(a) <= 5e-24)) {
		tmp = x + fma(t_0, 1.0, -tan(a));
	} else {
		tmp = x - (a + (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) <= -0.005) || !(tan(a) <= 5e-24))
		tmp = Float64(x + fma(t_0, 1.0, Float64(-tan(a))));
	else
		tmp = Float64(x - Float64(a + 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], -0.005], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-24]], $MachinePrecision]], N[(x + N[(t$95$0 * 1.0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(x - N[(a + N[(t$95$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) < -0.0050000000000000001 or 4.9999999999999998e-24 < (tan.f64 a)

   1. Initial program 80.1%

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

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

      3. fmm-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 80.6%

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

   if -0.0050000000000000001 < (tan.f64 a) < 4.9999999999999998e-24

   1. Initial program 77.5%

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

   3. Taylor expanded in a around 0 77.5%

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

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 5 \cdot 10^{-24}\right):\\ \;\;\;\;x + \mathsf{fma}\left(\tan y + \tan z, 1, -\tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

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

   3. fmm-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{-1 + \tan y \cdot \tan z}, -\tan a\right) \]
6. 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 78.9%

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

   1. tan-sum 50.0%

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

   2. div-inv 50.0%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 50.0%

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

   2. *-rgt-identity 50.0%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-6} \lor \neg \left(\tan a \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -1e-6) (not (<= (tan a) 2e-5)))
   (+ x (- (tan y) (tan a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -1e-6) || !(tan(a) <= 2e-5)) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = tan((y + z)) + (x - 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 ((tan(a) <= (-1d-6)) .or. (.not. (tan(a) <= 2d-5))) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = tan((y + z)) + (x - a)
    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) <= -1e-6) || !(Math.tan(a) <= 2e-5)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -1e-6) or not (math.tan(a) <= 2e-5):
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -1e-6) || !(tan(a) <= 2e-5))
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -1e-6) || ~((tan(a) <= 2e-5)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -1e-6], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 2e-5]], $MachinePrecision]], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -9.99999999999999955e-7 or 2.00000000000000016e-5 < (tan.f64 a)

   1. Initial program 78.7%

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

   3. Taylor expanded in z around 0 63.2%

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

      1. tan-quot 63.2%

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

      2. *-un-lft-identity 63.2%

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

   5. Applied egg-rr 63.2%

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

      1. *-lft-identity 63.2%

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

   7. Simplified 63.2%

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

   if -9.99999999999999955e-7 < (tan.f64 a) < 2.00000000000000016e-5

   1. Initial program 79.0%

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

   3. Taylor expanded in a around 0 79.0%

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

      1. associate-+r- 79.0%

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

   5. Applied egg-rr 79.0%

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

      1. associate-+r- 79.0%

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

      2. sub-neg 79.0%

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

      3. +-commutative 79.0%

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

      4. associate-+r+ 79.1%

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

      5. unsub-neg 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-6} \lor \neg \left(\tan a \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -1e-6)
   (+ x (- (tan y) (tan a)))
   (if (<= (tan a) 2e-5) (+ (tan (+ y z)) (- x a)) (+ (tan y) (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -1e-6) {
		tmp = x + (tan(y) - tan(a));
	} else if (tan(a) <= 2e-5) {
		tmp = tan((y + z)) + (x - a);
	} else {
		tmp = tan(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 (tan(a) <= (-1d-6)) then
        tmp = x + (tan(y) - tan(a))
    else if (tan(a) <= 2d-5) then
        tmp = tan((y + z)) + (x - a)
    else
        tmp = tan(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 (Math.tan(a) <= -1e-6) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else if (Math.tan(a) <= 2e-5) {
		tmp = Math.tan((y + z)) + (x - a);
	} else {
		tmp = Math.tan(y) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -1e-6:
		tmp = x + (math.tan(y) - math.tan(a))
	elif math.tan(a) <= 2e-5:
		tmp = math.tan((y + z)) + (x - a)
	else:
		tmp = math.tan(y) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -1e-6)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	elseif (tan(a) <= 2e-5)
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	else
		tmp = Float64(tan(y) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -1e-6)
		tmp = x + (tan(y) - tan(a));
	elseif (tan(a) <= 2e-5)
		tmp = tan((y + z)) + (x - a);
	else
		tmp = tan(y) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -1e-6], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-5], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(N[Tan[y], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -9.99999999999999955e-7

   1. Initial program 77.9%

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

   3. Taylor expanded in z around 0 60.7%

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

      1. tan-quot 60.7%

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

      2. *-un-lft-identity 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. *-lft-identity 60.7%

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

   7. Simplified 60.7%

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

   if -9.99999999999999955e-7 < (tan.f64 a) < 2.00000000000000016e-5

   1. Initial program 79.0%

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

   3. Taylor expanded in a around 0 79.0%

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

      1. associate-+r- 79.0%

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

   5. Applied egg-rr 79.0%

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

      1. associate-+r- 79.0%

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

      2. sub-neg 79.0%

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

      3. +-commutative 79.0%

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

      4. associate-+r+ 79.1%

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

      5. unsub-neg 79.1%

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

   7. Simplified 79.1%

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

   if 2.00000000000000016e-5 < (tan.f64 a)

   1. Initial program 79.7%

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

   3. Taylor expanded in z around 0 66.0%

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

      1. add-exp-log 57.3%

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

      2. tan-quot 57.3%

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

   5. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{e^{\log \left(x + \left(\tan y - \tan a\right)\right)}} \]
   6. Step-by-step derivation

      1. rem-exp-log 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-+l- 65.9%

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

   7. Applied egg-rr 65.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

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

   3. fmm-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0 79.4%

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

6. Final simplification 79.4%

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

Alternative 8: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 2e-17)
   (+ x (- (tan y) (tan a)))
   (+ x (+ (tan a) (tan (+ y z))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 2e-17) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(a) + 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) <= 2d-17) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(a) + 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) <= 2e-17) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(a) + Math.tan((y + z)));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 2e-17:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(a) + math.tan((y + z)))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 2e-17)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(a) + tan(Float64(y + z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 2e-17)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(a) + tan((y + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 2e-17], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[a], $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 2.00000000000000014e-17

   1. Initial program 82.8%

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

   3. Taylor expanded in z around 0 66.8%

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

      1. tan-quot 66.8%

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

      2. *-un-lft-identity 66.8%

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

   5. Applied egg-rr 66.8%

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

      1. *-lft-identity 66.8%

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

   7. Simplified 66.8%

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

   if 2.00000000000000014e-17 < (+.f64 y z)

   1. Initial program 73.9%

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

      1. sub-neg 73.9%

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

   4. Applied egg-rr 73.9%

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

      1. +-commutative 73.9%

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

      2. rem-square-sqrt 43.8%

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

      3. fabs-sqr 43.8%

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

      4. rem-square-sqrt 61.5%

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

      5. fabs-neg 61.5%

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

      6. rem-square-sqrt 17.7%

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

      7. fabs-sqr 17.7%

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

      8. rem-square-sqrt 42.5%

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

   6. Simplified 42.5%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan a + \tan \left(y + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 78.9%

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

3. Final simplification 78.9%

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

Alternative 10: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.65 \lor \neg \left(a \leq 16\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.65) (not (<= a 16.0)))
   (+ x (- y (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.65) || !(a <= 16.0)) {
		tmp = x + (y - tan(a));
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.65d0)) .or. (.not. (a <= 16.0d0))) then
        tmp = x + (y - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.65) || !(a <= 16.0)) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.65) or not (a <= 16.0):
		tmp = x + (y - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.65) || !(a <= 16.0))
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.65) || ~((a <= 16.0)))
		tmp = x + (y - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.65], N[Not[LessEqual[a, 16.0]], $MachinePrecision]], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.650000000000000022 or 16 < a

   1. Initial program 78.9%

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

   3. Taylor expanded in z around 0 63.0%

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

   4. Taylor expanded in y around 0 30.3%

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

   if -0.650000000000000022 < a < 16

   1. Initial program 78.9%

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

   3. Taylor expanded in a around 0 77.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.65 \lor \neg \left(a \leq 16\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.21 \lor \neg \left(a \leq 16\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.21) (not (<= a 16.0)))
   (+ x (- y (tan a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.21) || !(a <= 16.0)) {
		tmp = x + (y - tan(a));
	} else {
		tmp = tan((y + z)) + (x - 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 ((a <= (-0.21d0)) .or. (.not. (a <= 16.0d0))) then
        tmp = x + (y - tan(a))
    else
        tmp = tan((y + z)) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.21) || !(a <= 16.0)) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -0.21) or not (a <= 16.0):
		tmp = x + (y - math.tan(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.21) || !(a <= 16.0))
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -0.21) || ~((a <= 16.0)))
		tmp = x + (y - tan(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.21], N[Not[LessEqual[a, 16.0]], $MachinePrecision]], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.209999999999999992 or 16 < a

   1. Initial program 78.9%

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

   3. Taylor expanded in z around 0 63.0%

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

   4. Taylor expanded in y around 0 30.3%

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

   if -0.209999999999999992 < a < 16

   1. Initial program 78.9%

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

   3. Taylor expanded in a around 0 77.9%

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

      1. associate-+r- 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. associate-+r- 77.9%

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

      2. sub-neg 77.9%

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

      3. +-commutative 77.9%

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

      4. associate-+r+ 77.9%

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

      5. unsub-neg 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.21 \lor \neg \left(a \leq 16\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -0.00075) x (if (<= y 2.8e-9) (+ x (- y (tan a))) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.00075) {
		tmp = x;
	} else if (y <= 2.8e-9) {
		tmp = x + (y - tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-0.00075d0)) then
        tmp = x
    else if (y <= 2.8d-9) then
        tmp = x + (y - tan(a))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -0.00075) {
		tmp = x;
	} else if (y <= 2.8e-9) {
		tmp = x + (y - Math.tan(a));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -0.00075:
		tmp = x
	elif y <= 2.8e-9:
		tmp = x + (y - math.tan(a))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -0.00075)
		tmp = x;
	elseif (y <= 2.8e-9)
		tmp = Float64(x + Float64(y - tan(a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -0.00075)
		tmp = x;
	elseif (y <= 2.8e-9)
		tmp = x + (y - tan(a));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -0.00075], x, If[LessEqual[y, 2.8e-9], N[(x + N[(y - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000002e-4 or 2.79999999999999984e-9 < y

   1. Initial program 59.4%

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

   3. Taylor expanded in x around inf 24.0%

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

   if -7.5000000000000002e-4 < y < 2.79999999999999984e-9

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0 58.6%

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

   4. Taylor expanded in y around 0 58.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.7% 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 78.9%

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

3. Taylor expanded in x around inf 31.3%

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

4. Final simplification 31.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))