tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 30.7s

Alternatives: 14

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: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

Julia

function code(x, y, z, a)
	return Float64(x - Float64(tan(a) + Float64(Float64(tan(y) + tan(z)) / fma(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[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

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

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/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

   1. expm1-log1p-u 92.0%

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

   2. expm1-undefine 92.0%

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

   3. log1p-undefine 92.0%

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

   4. add-exp-log 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate--l+ 99.8%

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

   2. fma-neg 99.8%

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

   3. metadata-eval 99.8%

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

10. Simplified 99.8%

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

    1. sub-neg 99.8%

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

    2. associate--r+ 99.8%

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

    3. metadata-eval 99.8%

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

12. Applied egg-rr 99.8%

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

    1. sub-neg 99.8%

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

    2. sub0-neg 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.0001:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.0001)
     (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a)))
     (if (<= (tan a) 2e-23)
       (+ (/ t_0 (- 1.0 (* (tan y) (tan z)))) (- x a))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.0001) {
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	} else if (tan(a) <= 2e-23) {
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) + (x - a);
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (tan(a) <= (-0.0001d0)) then
        tmp = x + ((1.0d0 / (cos((y + z)) / sin((y + z)))) - tan(a))
    else if (tan(a) <= 2d-23) then
        tmp = (t_0 / (1.0d0 - (tan(y) * tan(z)))) + (x - a)
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (Math.tan(a) <= -0.0001) {
		tmp = x + ((1.0 / (Math.cos((y + z)) / Math.sin((y + z)))) - Math.tan(a));
	} else if (Math.tan(a) <= 2e-23) {
		tmp = (t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -0.0001:
		tmp = x + ((1.0 / (math.cos((y + z)) / math.sin((y + z)))) - math.tan(a))
	elif math.tan(a) <= 2e-23:
		tmp = (t_0 / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.0001)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)));
	elseif (tan(a) <= 2e-23)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -0.0001)
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	elseif (tan(a) <= 2e-23)
		tmp = (t_0 / (1.0 - (tan(y) * tan(z)))) + (x - a);
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.0001], N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-23], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -1.00000000000000005e-4

   1. Initial program 77.5%

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

      1. tan-quot 77.6%

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

      2. clear-num 77.6%

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

   4. Applied egg-rr 77.6%

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

   if -1.00000000000000005e-4 < (tan.f64 a) < 1.99999999999999992e-23

   1. Initial program 76.5%

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

      1. add-cbrt-cube 75.8%

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

      2. pow1/3 73.0%

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

      3. pow3 73.0%

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

      4. +-commutative 73.0%

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

      5. associate-+l- 73.0%

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

   4. Applied egg-rr 73.0%

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

      1. unpow1/3 75.9%

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

      2. rem-cbrt-cube 76.5%

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

      3. tan-sum 99.8%

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

      4. div-inv 99.8%

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

      5. fma-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around 0 99.8%

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

   if 1.99999999999999992e-23 < (tan.f64 a)

   1. Initial program 76.8%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   6. Simplified 99.7%

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

      1. expm1-log1p-u 94.9%

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

      2. expm1-undefine 94.9%

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

      3. log1p-undefine 94.8%

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

      4. add-exp-log 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate--l+ 99.7%

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

      2. fma-neg 99.7%

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

      3. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 77.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.0001:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -1e-8)
     (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a)))
     (if (<= (tan a) 2e-23)
       (+ x (/ t_0 (- 1.0 (* (tan y) (tan z)))))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -1e-8) {
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	} else if (tan(a) <= 2e-23) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (tan(a) <= (-1d-8)) then
        tmp = x + ((1.0d0 / (cos((y + z)) / sin((y + z)))) - tan(a))
    else if (tan(a) <= 2d-23) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (Math.tan(a) <= -1e-8) {
		tmp = x + ((1.0 / (Math.cos((y + z)) / Math.sin((y + z)))) - Math.tan(a));
	} else if (Math.tan(a) <= 2e-23) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -1e-8:
		tmp = x + ((1.0 / (math.cos((y + z)) / math.sin((y + z)))) - math.tan(a))
	elif math.tan(a) <= 2e-23:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -1e-8)
		tmp = Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)));
	elseif (tan(a) <= 2e-23)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -1e-8)
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	elseif (tan(a) <= 2e-23)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-8], N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-23], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -1e-8

   1. Initial program 77.9%

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

      1. tan-quot 77.9%

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

      2. clear-num 77.9%

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

   4. Applied egg-rr 77.9%

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

   if -1e-8 < (tan.f64 a) < 1.99999999999999992e-23

   1. Initial program 76.3%

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

      1. add-cbrt-cube 75.6%

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

      2. pow1/3 72.8%

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

      3. pow3 72.8%

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

      4. +-commutative 72.8%

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

      5. associate-+l- 72.8%

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

   4. Applied egg-rr 72.8%

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

      1. unpow1/3 75.6%

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

      2. rem-cbrt-cube 76.3%

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

      3. tan-sum 99.8%

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

      4. div-inv 99.8%

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

      5. fma-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around 0 99.8%

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

       1. neg-mul-1 99.8%

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

   11. Simplified 99.8%

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

   if 1.99999999999999992e-23 < (tan.f64 a)

   1. Initial program 76.8%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   6. Simplified 99.7%

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

      1. expm1-log1p-u 94.9%

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

      2. expm1-undefine 94.9%

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

      3. log1p-undefine 94.8%

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

      4. add-exp-log 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate--l+ 99.7%

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

      2. fma-neg 99.7%

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

      3. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 77.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

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

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/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

Alternative 5: 49.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.05)
   x
   (if (<= (tan a) 5e-10)
     (+ x (- (tan (+ y z)) a))
     (pow (pow x 3.0) 0.3333333333333333))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = x;
	} else if (tan(a) <= 5e-10) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = pow(pow(x, 3.0), 0.3333333333333333);
	}
	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) <= (-0.05d0)) then
        tmp = x
    else if (tan(a) <= 5d-10) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = (x ** 3.0d0) ** 0.3333333333333333d0
    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) <= -0.05) {
		tmp = x;
	} else if (Math.tan(a) <= 5e-10) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = Math.pow(Math.pow(x, 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.05:
		tmp = x
	elif math.tan(a) <= 5e-10:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = math.pow(math.pow(x, 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-10)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = (x ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.05)
		tmp = x;
	elseif (tan(a) <= 5e-10)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = (x ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], x, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-10], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[x, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.050000000000000003

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 24.4%

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

   if -0.050000000000000003 < (tan.f64 a) < 5.00000000000000031e-10

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0 77.1%

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

   if 5.00000000000000031e-10 < (tan.f64 a)

   1. Initial program 75.9%

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

      1. add-cbrt-cube 75.7%

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

      2. pow1/3 70.9%

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

      3. pow3 70.9%

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

      4. +-commutative 70.9%

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

      5. associate-+l- 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in x around inf 23.4%

      \[\leadsto {\color{blue}{\left({x}^{3}\right)}}^{0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

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

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/ 99.8%

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

   2. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

   1. expm1-log1p-u 92.0%

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

   2. expm1-undefine 92.0%

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

   3. log1p-undefine 92.0%

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

   4. add-exp-log 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate--l+ 99.8%

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

   2. fma-neg 99.8%

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

   3. metadata-eval 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in y around 0 77.0%

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

12. Final simplification 77.0%

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

Alternative 7: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-13} \lor \neg \left(a \leq 0.000425\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.6e-13) (not (<= a 0.000425)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.6e-13) || !(a <= 0.000425)) {
		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 ((a <= (-1.6d-13)) .or. (.not. (a <= 0.000425d0))) 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 ((a <= -1.6e-13) || !(a <= 0.000425)) {
		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 (a <= -1.6e-13) or not (a <= 0.000425):
		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 ((a <= -1.6e-13) || !(a <= 0.000425))
		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 ((a <= -1.6e-13) || ~((a <= 0.000425)))
		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[a, -1.6e-13], N[Not[LessEqual[a, 0.000425]], $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 a < -1.6e-13 or 4.24999999999999976e-4 < a

   1. Initial program 76.4%

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

   3. Taylor expanded in y around inf 57.5%

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

   if -1.6e-13 < a < 4.24999999999999976e-4

   1. Initial program 77.4%

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

   3. Taylor expanded in a around 0 77.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-13} \lor \neg \left(a \leq 0.000425\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -4e-13) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -4e-13) {
		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 ((y + z) <= (-4d-13)) 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 ((y + z) <= -4e-13) {
		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 (y + z) <= -4e-13:
		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 (Float64(y + z) <= -4e-13)
		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 ((y + z) <= -4e-13)
		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[N[(y + z), $MachinePrecision], -4e-13], 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 (+.f64 y z) < -4.0000000000000001e-13

   1. Initial program 64.3%

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

   3. Taylor expanded in y around inf 40.2%

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

   if -4.0000000000000001e-13 < (+.f64 y z)

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0 68.4%

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

Alternative 9: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

Alternative 10: 40.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -2.45 \cdot 10^{-294}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{elif}\;a \leq 1.58:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.35)
   x
   (if (<= a -2.45e-294)
     (+ x (- (tan y) a))
     (if (<= a 1.58) (+ x (- (tan z) a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.35) {
		tmp = x;
	} else if (a <= -2.45e-294) {
		tmp = x + (tan(y) - a);
	} else if (a <= 1.58) {
		tmp = x + (tan(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 (a <= (-1.35d0)) then
        tmp = x
    else if (a <= (-2.45d-294)) then
        tmp = x + (tan(y) - a)
    else if (a <= 1.58d0) then
        tmp = x + (tan(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 (a <= -1.35) {
		tmp = x;
	} else if (a <= -2.45e-294) {
		tmp = x + (Math.tan(y) - a);
	} else if (a <= 1.58) {
		tmp = x + (Math.tan(z) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.35:
		tmp = x
	elif a <= -2.45e-294:
		tmp = x + (math.tan(y) - a)
	elif a <= 1.58:
		tmp = x + (math.tan(z) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.35)
		tmp = x;
	elseif (a <= -2.45e-294)
		tmp = Float64(x + Float64(tan(y) - a));
	elseif (a <= 1.58)
		tmp = Float64(x + Float64(tan(z) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.35)
		tmp = x;
	elseif (a <= -2.45e-294)
		tmp = x + (tan(y) - a);
	elseif (a <= 1.58)
		tmp = x + (tan(z) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.35], x, If[LessEqual[a, -2.45e-294], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.58], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3500000000000001 or 1.5800000000000001 < a

   1. Initial program 76.5%

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

   3. Taylor expanded in x around inf 23.8%

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

   if -1.3500000000000001 < a < -2.4499999999999999e-294

   1. Initial program 76.5%

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

   3. Taylor expanded in a around 0 76.1%

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

   4. Taylor expanded in y around inf 60.2%

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

   if -2.4499999999999999e-294 < a < 1.5800000000000001

   1. Initial program 78.1%

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

   3. Taylor expanded in a around 0 78.1%

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

   4. Taylor expanded in y around 0 65.3%

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

Alternative 11: 50.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.58:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.55) x (if (<= a 1.58) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.55) {
		tmp = x;
	} else if (a <= 1.58) {
		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 (a <= (-1.55d0)) then
        tmp = x
    else if (a <= 1.58d0) 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 (a <= -1.55) {
		tmp = x;
	} else if (a <= 1.58) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.55:
		tmp = x
	elif a <= 1.58:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.55)
		tmp = x;
	elseif (a <= 1.58)
		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 (a <= -1.55)
		tmp = x;
	elseif (a <= 1.58)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.55], x, If[LessEqual[a, 1.58], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 1.5800000000000001 < a

   1. Initial program 76.5%

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

   3. Taylor expanded in x around inf 23.8%

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

   if -1.55000000000000004 < a < 1.5800000000000001

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0 77.1%

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

Alternative 12: 40.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.85) x (if (<= a 3.3e-10) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.85) {
		tmp = x;
	} else if (a <= 3.3e-10) {
		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 (a <= (-1.85d0)) then
        tmp = x
    else if (a <= 3.3d-10) 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 (a <= -1.85) {
		tmp = x;
	} else if (a <= 3.3e-10) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.85:
		tmp = x
	elif a <= 3.3e-10:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.85)
		tmp = x;
	elseif (a <= 3.3e-10)
		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 (a <= -1.85)
		tmp = x;
	elseif (a <= 3.3e-10)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.85], x, If[LessEqual[a, 3.3e-10], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8500000000000001 or 3.3e-10 < a

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf 23.9%

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

   if -1.8500000000000001 < a < 3.3e-10

   1. Initial program 77.1%

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

   3. Taylor expanded in a around 0 76.9%

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

   4. Taylor expanded in y around inf 59.7%

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

Alternative 13: 31.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 76.8%

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

3. Taylor expanded in x around inf 33.8%

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

Alternative 14: 3.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double a) {
	return 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 = a
end function

Java

public static double code(double x, double y, double z, double a) {
	return a;
}

Python

def code(x, y, z, a):
	return a

Julia

function code(x, y, z, a)
	return a
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := a

Derivation

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 36.1%

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

4. Taylor expanded in a around inf 3.4%

   \[\leadsto \color{blue}{-1 \cdot a} \]
5. Step-by-step derivation

   1. neg-mul-1 3.4%

      \[\leadsto \color{blue}{-a} \]

6. Simplified 3.4%

   \[\leadsto \color{blue}{-a} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 2.5%

      \[\leadsto \color{blue}{\sqrt{-a} \cdot \sqrt{-a}} \]

   2. sqrt-unprod 4.7%

      \[\leadsto \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}} \]

   3. sqr-neg 4.7%

      \[\leadsto \sqrt{\color{blue}{a \cdot a}} \]

   4. sqrt-unprod 2.5%

      \[\leadsto \color{blue}{\sqrt{a} \cdot \sqrt{a}} \]

   5. add-sqr-sqrt 3.3%

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

   6. *-un-lft-identity 3.3%

      \[\leadsto \color{blue}{1 \cdot a} \]

8. Applied egg-rr 3.3%

   \[\leadsto \color{blue}{1 \cdot a} \]
9. Step-by-step derivation

   1. *-lft-identity 3.3%

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

10. Simplified 3.3%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))