tan-example (used to crash)

Percentage Accurate: 78.8% → 99.6%

Time: 38.4s

Alternatives: 16

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: 78.8% 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, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return (((pow(tan(y), 2.0) / (tan(y) - tan(z))) + (pow(tan(z), 2.0) / (tan(z) - tan(y)))) / (1.0 - (tan(y) * tan(z)))) + (x - 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 = ((((tan(y) ** 2.0d0) / (tan(y) - tan(z))) + ((tan(z) ** 2.0d0) / (tan(z) - tan(y)))) / (1.0d0 - (tan(y) * tan(z)))) + (x - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return (((Math.pow(Math.tan(y), 2.0) / (Math.tan(y) - Math.tan(z))) + (Math.pow(Math.tan(z), 2.0) / (Math.tan(z) - Math.tan(y)))) / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - Math.tan(a));
}

Python

def code(x, y, z, a):
	return (((math.pow(math.tan(y), 2.0) / (math.tan(y) - math.tan(z))) + (math.pow(math.tan(z), 2.0) / (math.tan(z) - math.tan(y)))) / (1.0 - (math.tan(y) * math.tan(z)))) + (x - math.tan(a))

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	tmp = ((((tan(y) ^ 2.0) / (tan(y) - tan(z))) + ((tan(z) ^ 2.0) / (tan(z) - tan(y)))) / (1.0 - (tan(y) * tan(z)))) + (x - tan(a));
end

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. +-commutative 81.1%

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

   2. associate-+l- 81.1%

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

   3. tan-sum 99.7%

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

   4. div-inv 99.6%

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

   5. fmm-def 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fmm-undef 99.6%

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

   2. add-cube-cbrt 97.9%

      \[\leadsto \left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \color{blue}{\left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right) \cdot \sqrt[3]{\tan a - x}} \]

   3. prod-diff 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right)} \]

   4. pow2 97.9%

      \[\leadsto \mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot \color{blue}{{\left(\sqrt[3]{\tan a - x}\right)}^{2}}\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right) \]

6. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot {\left(\sqrt[3]{\tan a - x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, {\left(\sqrt[3]{\tan a - x}\right)}^{2}, \sqrt[3]{\tan a - x} \cdot {\left(\sqrt[3]{\tan a - x}\right)}^{2}\right)} \]
7. Step-by-step derivation

8. Simplified 99.7%

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

   1. flip-+ 99.6%

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

   2. div-sub 99.7%

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

   3. pow2 99.7%

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

   4. pow2 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in x around 0 99.7%

    \[\leadsto \left(\frac{\frac{{\tan y}^{2}}{\tan y - \tan z} - \frac{{\tan z}^{2}}{\tan y - \tan z}}{1 - \tan y \cdot \tan z} + \left(x - \tan a\right)\right) + \color{blue}{0} \]

12. Final simplification 99.7%

    \[\leadsto \frac{\frac{{\tan y}^{2}}{\tan y - \tan z} + \frac{{\tan z}^{2}}{\tan z - \tan y}}{1 - \tan y \cdot \tan z} + \left(x - \tan a\right) \]
13. Add Preprocessing

Alternative 2: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{-1}{\tan y \cdot \tan z + -1} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 5e-51)
     (+ x (- (* (+ (tan y) (tan z)) (/ -1.0 (+ (* (tan y) (tan z)) -1.0))) a))
     (+ (/ (sin (+ y z)) (cos (+ y z))) (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 5e-51) {
		tmp = x + (((tan(y) + tan(z)) * (-1.0 / ((tan(y) * tan(z)) + -1.0))) - a);
	} else {
		tmp = (sin((y + z)) / cos((y + 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 (tan(a) <= (-0.01d0)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 5d-51) then
        tmp = x + (((tan(y) + tan(z)) * ((-1.0d0) / ((tan(y) * tan(z)) + (-1.0d0)))) - a)
    else
        tmp = (sin((y + z)) / cos((y + 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 (Math.tan(a) <= -0.01) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-51) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) * (-1.0 / ((Math.tan(y) * Math.tan(z)) + -1.0))) - a);
	} else {
		tmp = (Math.sin((y + z)) / Math.cos((y + z))) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 5e-51:
		tmp = x + (((math.tan(y) + math.tan(z)) * (-1.0 / ((math.tan(y) * math.tan(z)) + -1.0))) - a)
	else:
		tmp = (math.sin((y + z)) / math.cos((y + z))) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 5e-51)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) * Float64(-1.0 / Float64(Float64(tan(y) * tan(z)) + -1.0))) - a));
	else
		tmp = Float64(Float64(sin(Float64(y + z)) / cos(Float64(y + z))) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 5e-51)
		tmp = x + (((tan(y) + tan(z)) * (-1.0 / ((tan(y) * tan(z)) + -1.0))) - a);
	else
		tmp = (sin((y + z)) / cos((y + z))) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-51], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 83.6%

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

   if -0.0100000000000000002 < (tan.f64 a) < 5.00000000000000004e-51

   1. Initial program 81.4%

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

   3. Taylor expanded in a around 0 81.4%

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

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

   if 5.00000000000000004e-51 < (tan.f64 a)

   1. Initial program 77.9%

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

      1. +-commutative 77.9%

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

      2. sub-neg 77.9%

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

      3. associate-+l+ 78.0%

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

      4. tan-quot 78.0%

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

      5. div-inv 77.9%

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

      6. fma-define 77.9%

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

      7. neg-mul-1 77.9%

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

      8. fma-define 77.9%

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

   4. Applied egg-rr 77.9%

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

      1. fma-undefine 77.9%

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

      2. neg-mul-1 77.9%

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

      3. neg-sub0 77.9%

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

      4. associate-+l- 77.9%

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

      5. neg-sub0 77.9%

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

      6. fmm-undef 77.9%

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

      7. associate-*r/ 78.0%

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

      8. *-rgt-identity 78.0%

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

   6. Simplified 78.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{-1}{\tan y \cdot \tan z + -1} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x - \left(a + \frac{\tan y + \tan z}{\tan y \cdot \tan z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 5e-51)
     (- x (+ a (/ (+ (tan y) (tan z)) (+ (* (tan y) (tan z)) -1.0))))
     (+ (/ (sin (+ y z)) (cos (+ y z))) (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 5e-51) {
		tmp = x - (a + ((tan(y) + tan(z)) / ((tan(y) * tan(z)) + -1.0)));
	} else {
		tmp = (sin((y + z)) / cos((y + 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 (tan(a) <= (-0.01d0)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 5d-51) then
        tmp = x - (a + ((tan(y) + tan(z)) / ((tan(y) * tan(z)) + (-1.0d0))))
    else
        tmp = (sin((y + z)) / cos((y + 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 (Math.tan(a) <= -0.01) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-51) {
		tmp = x - (a + ((Math.tan(y) + Math.tan(z)) / ((Math.tan(y) * Math.tan(z)) + -1.0)));
	} else {
		tmp = (Math.sin((y + z)) / Math.cos((y + z))) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 5e-51:
		tmp = x - (a + ((math.tan(y) + math.tan(z)) / ((math.tan(y) * math.tan(z)) + -1.0)))
	else:
		tmp = (math.sin((y + z)) / math.cos((y + z))) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 5e-51)
		tmp = Float64(x - Float64(a + Float64(Float64(tan(y) + tan(z)) / Float64(Float64(tan(y) * tan(z)) + -1.0))));
	else
		tmp = Float64(Float64(sin(Float64(y + z)) / cos(Float64(y + z))) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 5e-51)
		tmp = x - (a + ((tan(y) + tan(z)) / ((tan(y) * tan(z)) + -1.0)));
	else
		tmp = (sin((y + z)) / cos((y + z))) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-51], N[(x - N[(a + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 83.6%

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

   if -0.0100000000000000002 < (tan.f64 a) < 5.00000000000000004e-51

   1. Initial program 81.4%

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

   3. Taylor expanded in a around 0 81.4%

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

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

      3. fmm-def 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fmm-undef 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-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%

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

   if 5.00000000000000004e-51 < (tan.f64 a)

   1. Initial program 77.9%

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

      1. +-commutative 77.9%

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

      2. sub-neg 77.9%

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

      3. associate-+l+ 78.0%

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

      4. tan-quot 78.0%

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

      5. div-inv 77.9%

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

      6. fma-define 77.9%

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

      7. neg-mul-1 77.9%

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

      8. fma-define 77.9%

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

   4. Applied egg-rr 77.9%

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

      1. fma-undefine 77.9%

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

      2. neg-mul-1 77.9%

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

      3. neg-sub0 77.9%

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

      4. associate-+l- 77.9%

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

      5. neg-sub0 77.9%

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

      6. fmm-undef 77.9%

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

      7. associate-*r/ 78.0%

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

      8. *-rgt-identity 78.0%

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

   6. Simplified 78.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x - \left(a + \frac{\tan y + \tan z}{\tan y \cdot \tan z + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. +-commutative 81.1%

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

   2. associate-+l- 81.1%

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

   3. tan-sum 99.7%

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

   4. div-inv 99.6%

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

   5. fmm-def 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fmm-undef 99.6%

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

   2. add-cube-cbrt 97.9%

      \[\leadsto \left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \color{blue}{\left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right) \cdot \sqrt[3]{\tan a - x}} \]

   3. prod-diff 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right)} \]

   4. pow2 97.9%

      \[\leadsto \mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot \color{blue}{{\left(\sqrt[3]{\tan a - x}\right)}^{2}}\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}, \sqrt[3]{\tan a - x} \cdot \left(\sqrt[3]{\tan a - x} \cdot \sqrt[3]{\tan a - x}\right)\right) \]

6. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\sqrt[3]{\tan a - x} \cdot {\left(\sqrt[3]{\tan a - x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a - x}, {\left(\sqrt[3]{\tan a - x}\right)}^{2}, \sqrt[3]{\tan a - x} \cdot {\left(\sqrt[3]{\tan a - x}\right)}^{2}\right)} \]
7. Step-by-step derivation

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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)) / ((tan(y) * tan(z)) + (-1.0d0))))
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)) / ((Math.tan(y) * Math.tan(z)) + -1.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. +-commutative 81.1%

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

   2. sub-neg 81.1%

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

   3. associate-+l+ 81.1%

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

   4. tan-sum 99.7%

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

   5. div-inv 99.6%

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

   6. fma-define 99.7%

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

   7. neg-mul-1 99.7%

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

   8. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fma-undefine 99.6%

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

   2. fma-undefine 99.6%

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

   3. neg-mul-1 99.6%

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

   4. associate-+r+ 99.6%

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

   5. sub-neg 99.6%

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

   6. associate-*r/ 99.6%

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

   7. *-rgt-identity 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.01) (not (<= (tan a) 1e-12)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.01) || !(tan(a) <= 1e-12)) {
		tmp = x + (tan(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 ((tan(a) <= (-0.01d0)) .or. (.not. (tan(a) <= 1d-12))) then
        tmp = x + (tan(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 ((Math.tan(a) <= -0.01) || !(Math.tan(a) <= 1e-12)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.01) or not (math.tan(a) <= 1e-12):
		tmp = x + (math.tan(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 ((tan(a) <= -0.01) || !(tan(a) <= 1e-12))
		tmp = Float64(x + Float64(tan(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 ((tan(a) <= -0.01) || ~((tan(a) <= 1e-12)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-12]], $MachinePrecision]], N[(x + N[(N[Tan[y], $MachinePrecision] - 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 (tan.f64 a) < -0.0100000000000000002 or 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf 63.9%

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

   if -0.0100000000000000002 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   (exp (log x))
   (if (<= (tan a) 1e-12) (+ x (- (tan (+ y z)) a)) (expm1 (log1p x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = exp(log(x));
	} else if (tan(a) <= 1e-12) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = expm1(log1p(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (Math.tan(a) <= -0.01) {
		tmp = Math.exp(Math.log(x));
	} else if (Math.tan(a) <= 1e-12) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = Math.expm1(Math.log1p(x));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = math.exp(math.log(x))
	elif math.tan(a) <= 1e-12:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = math.expm1(math.log1p(x))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = exp(log(x));
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = expm1(log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + x], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 83.6%

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

      1. add-cbrt-cube 83.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      2. pow1/3 79.7%

         \[\leadsto \color{blue}{{\left(\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 79.8%

         \[\leadsto {\color{blue}{\left({\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 79.8%

         \[\leadsto {\left({\color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. associate-+l- 79.8%

         \[\leadsto {\left({\color{blue}{\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. add-exp-log 79.9%

         \[\leadsto {\color{blue}{\left(e^{\log \left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}\right)}}^{0.3333333333333333} \]

      2. log-pow 79.6%

         \[\leadsto {\left(e^{\color{blue}{3 \cdot \log \left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}\right)}^{0.3333333333333333} \]

      3. associate--r- 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}\right)}^{0.3333333333333333} \]

      4. pow1 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{{\tan \left(y + z\right)}^{1}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      5. metadata-eval 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left({\tan \left(y + z\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      6. sqrt-pow1 65.5%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{\sqrt{{\tan \left(y + z\right)}^{2}}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      7. +-commutative 65.5%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(x + \left(\sqrt{{\tan \left(y + z\right)}^{2}} - \tan a\right)\right)}}\right)}^{0.3333333333333333} \]

      8. sqrt-pow1 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{{\tan \left(y + z\right)}^{\left(\frac{2}{2}\right)}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left({\tan \left(y + z\right)}^{\color{blue}{1}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      10. pow1 79.6%

          \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 79.6%

      \[\leadsto {\color{blue}{\left(e^{3 \cdot \log \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}\right)}}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 23.2%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. mul-1-neg 23.2%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]

      2. log-rec 23.2%

         \[\leadsto {\left(e^{3 \cdot \left(-\color{blue}{\left(-\log x\right)}\right)}\right)}^{0.3333333333333333} \]

      3. remove-double-neg 23.2%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]

   9. Simplified 23.2%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. *-commutative 23.2%

          \[\leadsto {\left(e^{\color{blue}{\log x \cdot 3}}\right)}^{0.3333333333333333} \]

       2. pow-to-exp 23.2%

          \[\leadsto {\color{blue}{\left({x}^{3}\right)}}^{0.3333333333333333} \]

       3. pow3 23.2%

          \[\leadsto {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{0.3333333333333333} \]

       4. pow1/3 23.2%

          \[\leadsto \color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \]

       5. add-cbrt-cube 23.2%

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

       6. add-exp-log 23.2%

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

   11. Applied egg-rr 23.2%

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

   if -0.0100000000000000002 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

   if 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 76.9%

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

      1. add-cbrt-cube 76.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      2. pow1/3 58.9%

         \[\leadsto \color{blue}{{\left(\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 58.9%

         \[\leadsto {\color{blue}{\left({\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 58.9%

         \[\leadsto {\left({\color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. associate-+l- 58.9%

         \[\leadsto {\left({\color{blue}{\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 58.9%

      \[\leadsto \color{blue}{{\left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. add-exp-log 59.1%

         \[\leadsto {\color{blue}{\left(e^{\log \left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}\right)}}^{0.3333333333333333} \]

      2. log-pow 58.9%

         \[\leadsto {\left(e^{\color{blue}{3 \cdot \log \left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}\right)}^{0.3333333333333333} \]

      3. associate--r- 58.8%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}\right)}^{0.3333333333333333} \]

      4. pow1 58.8%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{{\tan \left(y + z\right)}^{1}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      5. metadata-eval 58.8%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left({\tan \left(y + z\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      6. sqrt-pow1 41.7%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{\sqrt{{\tan \left(y + z\right)}^{2}}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      7. +-commutative 41.7%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(x + \left(\sqrt{{\tan \left(y + z\right)}^{2}} - \tan a\right)\right)}}\right)}^{0.3333333333333333} \]

      8. sqrt-pow1 58.8%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{{\tan \left(y + z\right)}^{\left(\frac{2}{2}\right)}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 58.8%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left({\tan \left(y + z\right)}^{\color{blue}{1}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      10. pow1 58.8%

          \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 58.8%

      \[\leadsto {\color{blue}{\left(e^{3 \cdot \log \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}\right)}}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 18.4%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. mul-1-neg 18.4%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]

      2. log-rec 18.4%

         \[\leadsto {\left(e^{3 \cdot \left(-\color{blue}{\left(-\log x\right)}\right)}\right)}^{0.3333333333333333} \]

      3. remove-double-neg 18.4%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]

   9. Simplified 18.4%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. *-commutative 18.4%

          \[\leadsto {\left(e^{\color{blue}{\log x \cdot 3}}\right)}^{0.3333333333333333} \]

       2. pow-to-exp 18.4%

          \[\leadsto {\color{blue}{\left({x}^{3}\right)}}^{0.3333333333333333} \]

       3. pow3 18.4%

          \[\leadsto {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{0.3333333333333333} \]

       4. pow1/3 18.4%

          \[\leadsto \color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \]

       5. expm1-log1p-u 18.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\left(x \cdot x\right) \cdot x}\right)\right)} \]

       6. add-cbrt-cube 18.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x}\right)\right) \]

       7. expm1-undefine 18.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]

   11. Applied egg-rr 18.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-define 18.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   13. Simplified 18.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return log(exp((tan((y + z)) + (x - 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 = log(exp((tan((y + z)) + (x - tan(a)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. add-log-exp 81.1%

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

   2. +-commutative 81.1%

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

   3. associate-+l- 81.1%

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

4. Applied egg-rr 81.1%

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

5. Final simplification 81.1%

   \[\leadsto \log \left(e^{\tan \left(y + z\right) + \left(x - \tan a\right)}\right) \]
6. Add Preprocessing

Alternative 9: 49.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;e^{\log x}\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   (exp (log x))
   (if (<= (tan a) 1e-12) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = exp(log(x));
	} else if (tan(a) <= 1e-12) {
		tmp = x + (tan((y + z)) - 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 (tan(a) <= (-0.01d0)) then
        tmp = exp(log(x))
    else if (tan(a) <= 1d-12) then
        tmp = x + (tan((y + z)) - 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 (Math.tan(a) <= -0.01) {
		tmp = Math.exp(Math.log(x));
	} else if (Math.tan(a) <= 1e-12) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = math.exp(math.log(x))
	elif math.tan(a) <= 1e-12:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = exp(log(x));
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = exp(log(x));
	elseif (tan(a) <= 1e-12)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 83.6%

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

      1. add-cbrt-cube 83.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      2. pow1/3 79.7%

         \[\leadsto \color{blue}{{\left(\left(\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right) \cdot \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 79.8%

         \[\leadsto {\color{blue}{\left({\left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 79.8%

         \[\leadsto {\left({\color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. associate-+l- 79.8%

         \[\leadsto {\left({\color{blue}{\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{{\left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. add-exp-log 79.9%

         \[\leadsto {\color{blue}{\left(e^{\log \left({\left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}^{3}\right)}\right)}}^{0.3333333333333333} \]

      2. log-pow 79.6%

         \[\leadsto {\left(e^{\color{blue}{3 \cdot \log \left(\tan \left(y + z\right) - \left(\tan a - x\right)\right)}}\right)}^{0.3333333333333333} \]

      3. associate--r- 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(\left(\tan \left(y + z\right) - \tan a\right) + x\right)}}\right)}^{0.3333333333333333} \]

      4. pow1 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{{\tan \left(y + z\right)}^{1}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      5. metadata-eval 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left({\tan \left(y + z\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      6. sqrt-pow1 65.5%

         \[\leadsto {\left(e^{3 \cdot \log \left(\left(\color{blue}{\sqrt{{\tan \left(y + z\right)}^{2}}} - \tan a\right) + x\right)}\right)}^{0.3333333333333333} \]

      7. +-commutative 65.5%

         \[\leadsto {\left(e^{3 \cdot \log \color{blue}{\left(x + \left(\sqrt{{\tan \left(y + z\right)}^{2}} - \tan a\right)\right)}}\right)}^{0.3333333333333333} \]

      8. sqrt-pow1 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{{\tan \left(y + z\right)}^{\left(\frac{2}{2}\right)}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 79.6%

         \[\leadsto {\left(e^{3 \cdot \log \left(x + \left({\tan \left(y + z\right)}^{\color{blue}{1}} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

      10. pow1 79.6%

          \[\leadsto {\left(e^{3 \cdot \log \left(x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right)\right)}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 79.6%

      \[\leadsto {\color{blue}{\left(e^{3 \cdot \log \left(x + \left(\tan \left(y + z\right) - \tan a\right)\right)}\right)}}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 23.2%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. mul-1-neg 23.2%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}}\right)}^{0.3333333333333333} \]

      2. log-rec 23.2%

         \[\leadsto {\left(e^{3 \cdot \left(-\color{blue}{\left(-\log x\right)}\right)}\right)}^{0.3333333333333333} \]

      3. remove-double-neg 23.2%

         \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]

   9. Simplified 23.2%

      \[\leadsto {\left(e^{3 \cdot \color{blue}{\log x}}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. *-commutative 23.2%

          \[\leadsto {\left(e^{\color{blue}{\log x \cdot 3}}\right)}^{0.3333333333333333} \]

       2. pow-to-exp 23.2%

          \[\leadsto {\color{blue}{\left({x}^{3}\right)}}^{0.3333333333333333} \]

       3. pow3 23.2%

          \[\leadsto {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{0.3333333333333333} \]

       4. pow1/3 23.2%

          \[\leadsto \color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \]

       5. add-cbrt-cube 23.2%

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

       6. add-exp-log 23.2%

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

   11. Applied egg-rr 23.2%

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

   if -0.0100000000000000002 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

   if 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 76.9%

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

   3. Taylor expanded in x around inf 18.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01) x (if (<= (tan a) 1e-12) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x;
	} else if (tan(a) <= 1e-12) {
		tmp = x + (tan((y + z)) - 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 (tan(a) <= (-0.01d0)) then
        tmp = x
    else if (tan(a) <= 1d-12) then
        tmp = x + (tan((y + z)) - 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 (Math.tan(a) <= -0.01) {
		tmp = x;
	} else if (Math.tan(a) <= 1e-12) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = x
	elif math.tan(a) <= 1e-12:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = x;
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = x;
	elseif (tan(a) <= 1e-12)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], x, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0100000000000000002 or 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 80.4%

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

   3. Taylor expanded in x around inf 20.8%

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

   if -0.0100000000000000002 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.9%

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

Alternative 11: 39.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-51}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -1e-5) x (if (<= (tan a) 5e-51) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -1e-5) {
		tmp = x;
	} else if (tan(a) <= 5e-51) {
		tmp = x + (tan(y) - 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 (tan(a) <= (-1d-5)) then
        tmp = x
    else if (tan(a) <= 5d-51) then
        tmp = x + (tan(y) - 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 (Math.tan(a) <= -1e-5) {
		tmp = x;
	} else if (Math.tan(a) <= 5e-51) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -1e-5:
		tmp = x
	elif math.tan(a) <= 5e-51:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -1e-5)
		tmp = x;
	elseif (tan(a) <= 5e-51)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -1e-5)
		tmp = x;
	elseif (tan(a) <= 5e-51)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -1e-5], x, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-51], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -1.00000000000000008e-5 or 5.00000000000000004e-51 < (tan.f64 a)

   1. Initial program 80.9%

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

   3. Taylor expanded in x around inf 20.8%

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

   if -1.00000000000000008e-5 < (tan.f64 a) < 5.00000000000000004e-51

   1. Initial program 81.2%

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

   3. Taylor expanded in a around 0 81.2%

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

   4. Taylor expanded in y around inf 59.4%

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

Alternative 12: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.55e-7) (- (+ (tan y) x) (tan a)) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.55e-7) {
		tmp = (tan(y) + x) - tan(a);
	} else {
		tmp = x + (tan(z) - 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 (z <= 1.55d-7) then
        tmp = (tan(y) + x) - tan(a)
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.55e-7) {
		tmp = (Math.tan(y) + x) - Math.tan(a);
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.55e-7:
		tmp = (math.tan(y) + x) - math.tan(a)
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.55e-7)
		tmp = Float64(Float64(tan(y) + x) - tan(a));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.55e-7)
		tmp = (tan(y) + x) - tan(a);
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.55e-7], N[(N[(N[Tan[y], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.55e-7

   1. Initial program 87.9%

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

   3. Taylor expanded in a around inf 87.9%

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

   4. Taylor expanded in y around inf 74.9%

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

      1. tan-quot 74.9%

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

      2. associate-+r- 74.9%

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

   6. Applied egg-rr 74.9%

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

   if 1.55e-7 < z

   1. Initial program 60.0%

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

   3. Taylor expanded in y around 0 60.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.8e-7) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.8e-7) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - 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 (z <= 1.8d-7) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.8e-7) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.8e-7:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.8e-7)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.8e-7)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.8e-7], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.79999999999999997e-7

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf 74.9%

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

   if 1.79999999999999997e-7 < z

   1. Initial program 60.0%

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

   3. Taylor expanded in y around 0 60.0%

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

Alternative 14: 78.8% 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 81.1%

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

Alternative 15: 33.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5000000000000:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{elif}\;y + z \leq 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \left(a - \tan z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5000000000000.0)
   (+ x (- (tan y) a))
   (if (<= (+ y z) 1e-16) x (- x (- a (tan z))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5000000000000.0) {
		tmp = x + (tan(y) - a);
	} else if ((y + z) <= 1e-16) {
		tmp = x;
	} else {
		tmp = x - (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 + z) <= (-5000000000000.0d0)) then
        tmp = x + (tan(y) - a)
    else if ((y + z) <= 1d-16) then
        tmp = x
    else
        tmp = x - (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 + z) <= -5000000000000.0) {
		tmp = x + (Math.tan(y) - a);
	} else if ((y + z) <= 1e-16) {
		tmp = x;
	} else {
		tmp = x - (a - Math.tan(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5000000000000.0:
		tmp = x + (math.tan(y) - a)
	elif (y + z) <= 1e-16:
		tmp = x
	else:
		tmp = x - (a - math.tan(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5000000000000.0)
		tmp = Float64(x + Float64(tan(y) - a));
	elseif (Float64(y + z) <= 1e-16)
		tmp = x;
	else
		tmp = Float64(x - Float64(a - tan(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5000000000000.0)
		tmp = x + (tan(y) - a);
	elseif ((y + z) <= 1e-16)
		tmp = x;
	else
		tmp = x - (a - tan(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5000000000000.0], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1e-16], x, N[(x - N[(a - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -5e12

   1. Initial program 72.4%

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

   3. Taylor expanded in a around 0 40.3%

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

   4. Taylor expanded in y around inf 25.4%

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

   if -5e12 < (+.f64 y z) < 9.9999999999999998e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 48.3%

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

   if 9.9999999999999998e-17 < (+.f64 y z)

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0 40.0%

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

   4. Taylor expanded in y around 0 24.4%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -5000000000000:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{elif}\;y + z \leq 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \left(a - \tan z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.2% 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 81.1%

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

3. Taylor expanded in x around inf 28.9%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))