tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 27.4s

Alternatives: 10

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: 79.8% 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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

5. Simplified 99.6%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
6. 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-udef 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-udef 94.9%

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

   4. add-exp-log 99.6%

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

7. Applied egg-rr 99.6%

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

   1. associate--l+ 99.6%

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

9. Applied egg-rr 99.6%

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

    1. fma-neg 99.6%

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

    2. metadata-eval 99.6%

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

11. Simplified 99.6%

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

    1. associate-+r- 99.5%

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

    2. associate--r+ 99.5%

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

    3. metadata-eval 99.5%

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

13. Applied egg-rr 99.5%

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

    1. associate--l+ 99.6%

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

    2. sub0-neg 99.6%

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

15. Simplified 99.6%

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

16. Final simplification 99.6%

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

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\log \left(\frac{1 + \mathsf{expm1}\left(x + t_0\right)}{e^{\tan a}}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -1e-11)
     (log (/ (+ 1.0 (expm1 (+ x t_0))) (exp (tan a))))
     (if (<= (tan a) 5e-20)
       (fma (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z)))) x)
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -1e-11) {
		tmp = log(((1.0 + expm1((x + t_0))) / exp(tan(a))));
	} else if (tan(a) <= 5e-20) {
		tmp = fma((tan(y) + tan(z)), (1.0 / (1.0 - (tan(y) * tan(z)))), x);
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -1e-11)
		tmp = log(Float64(Float64(1.0 + expm1(Float64(x + t_0))) / exp(tan(a))));
	elseif (tan(a) <= 5e-20)
		tmp = fma(Float64(tan(y) + tan(z)), Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), x);
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-11], N[Log[N[(N[(1.0 + N[(Exp[N[(x + t$95$0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[Exp[N[Tan[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-20], N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -9.99999999999999939e-12

   1. Initial program 73.8%

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

      1. associate-+r- 73.8%

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

      2. log1p-expm1-u 73.8%

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

      3. log1p-udef 73.8%

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

      4. add-log-exp 73.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)\right) - \color{blue}{\log \left(e^{\tan a}\right)} \]

      5. diff-log 73.9%

         \[\leadsto \color{blue}{\log \left(\frac{1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)}{e^{\tan a}}\right)} \]

      6. +-commutative 73.9%

         \[\leadsto \log \left(\frac{1 + \mathsf{expm1}\left(\color{blue}{\tan \left(y + z\right) + x}\right)}{e^{\tan a}}\right) \]

   3. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\log \left(\frac{1 + \mathsf{expm1}\left(\tan \left(y + z\right) + x\right)}{e^{\tan a}}\right)} \]

   if -9.99999999999999939e-12 < (tan.f64 a) < 4.9999999999999999e-20

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-+l- 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in a around 0 80.0%

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

      1. neg-mul-1 80.0%

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

   6. Simplified 80.0%

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

      1. sub-neg 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. remove-double-neg 80.0%

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

      2. +-commutative 80.0%

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

      3. +-commutative 80.0%

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

   10. Simplified 80.0%

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

       1. +-commutative 80.0%

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

       2. +-commutative 80.0%

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

       3. tan-sum 99.6%

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

       4. div-inv 99.6%

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

       5. fma-def 99.6%

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

   12. Applied egg-rr 99.6%

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

   if 4.9999999999999999e-20 < (tan.f64 a)

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\log \left(\frac{1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)}{e^{\tan a}}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\log \left(\frac{1 + \mathsf{expm1}\left(x + t_0\right)}{e^{\tan a}}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \left(-1 + \left(1 + \tan y \cdot \tan z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -1e-11)
     (log (/ (+ 1.0 (expm1 (+ x t_0))) (exp (tan a))))
     (if (<= (tan a) 5e-20)
       (+
        x
        (/ (+ (tan y) (tan z)) (- 1.0 (+ -1.0 (+ 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 + z));
	double tmp;
	if (tan(a) <= -1e-11) {
		tmp = log(((1.0 + expm1((x + t_0))) / exp(tan(a))));
	} else if (tan(a) <= 5e-20) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (-1.0 + (1.0 + (tan(y) * tan(z))))));
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (Math.tan(a) <= -1e-11) {
		tmp = Math.log(((1.0 + Math.expm1((x + t_0))) / Math.exp(Math.tan(a))));
	} else if (Math.tan(a) <= 5e-20) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (-1.0 + (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 + z))
	tmp = 0
	if math.tan(a) <= -1e-11:
		tmp = math.log(((1.0 + math.expm1((x + t_0))) / math.exp(math.tan(a))))
	elif math.tan(a) <= 5e-20:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (-1.0 + (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 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -1e-11)
		tmp = log(Float64(Float64(1.0 + expm1(Float64(x + t_0))) / exp(tan(a))));
	elseif (tan(a) <= 5e-20)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(-1.0 + Float64(1.0 + Float64(tan(y) * tan(z)))))));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-11], N[Log[N[(N[(1.0 + N[(Exp[N[(x + t$95$0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[Exp[N[Tan[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-20], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-1.0 + N[(1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $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) < -9.99999999999999939e-12

   1. Initial program 73.8%

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

      1. associate-+r- 73.8%

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

      2. log1p-expm1-u 73.8%

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

      3. log1p-udef 73.8%

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

      4. add-log-exp 73.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)\right) - \color{blue}{\log \left(e^{\tan a}\right)} \]

      5. diff-log 73.9%

         \[\leadsto \color{blue}{\log \left(\frac{1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)}{e^{\tan a}}\right)} \]

      6. +-commutative 73.9%

         \[\leadsto \log \left(\frac{1 + \mathsf{expm1}\left(\color{blue}{\tan \left(y + z\right) + x}\right)}{e^{\tan a}}\right) \]

   3. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\log \left(\frac{1 + \mathsf{expm1}\left(\tan \left(y + z\right) + x\right)}{e^{\tan a}}\right)} \]

   if -9.99999999999999939e-12 < (tan.f64 a) < 4.9999999999999999e-20

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-+l- 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in a around 0 80.0%

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

      1. neg-mul-1 80.0%

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

   6. Simplified 80.0%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. 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%

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

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \left(-x\right) \]
   9. 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%

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

   10. Simplified 99.6%

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

       1. expm1-log1p-u 95.0%

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

       2. expm1-udef 95.0%

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

       3. log1p-udef 95.0%

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

       4. add-exp-log 99.7%

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

   12. Applied egg-rr 99.6%

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

   if 4.9999999999999999e-20 < (tan.f64 a)

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\log \left(\frac{1 + \mathsf{expm1}\left(x + \tan \left(y + z\right)\right)}{e^{\tan a}}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \left(-1 + \left(1 + \tan y \cdot \tan z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

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 79.5%

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

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \left(-1 + \left(1 + \tan y \cdot \tan z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0 + \left(x - \tan a\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -3.5e-12)
     (+ x (- t_0 (tan a)))
     (if (<= a 1.25e-17)
       (+
        x
        (/ (+ (tan y) (tan z)) (- 1.0 (+ -1.0 (+ 1.0 (* (tan y) (tan z)))))))
       (log (exp (+ t_0 (- x (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -3.5e-12) {
		tmp = x + (t_0 - tan(a));
	} else if (a <= 1.25e-17) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (-1.0 + (1.0 + (tan(y) * tan(z))))));
	} else {
		tmp = log(exp((t_0 + (x - tan(a)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-3.5d-12)) then
        tmp = x + (t_0 - tan(a))
    else if (a <= 1.25d-17) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - ((-1.0d0) + (1.0d0 + (tan(y) * tan(z))))))
    else
        tmp = log(exp((t_0 + (x - tan(a)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -3.5e-12) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (a <= 1.25e-17) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (-1.0 + (1.0 + (Math.tan(y) * Math.tan(z))))));
	} else {
		tmp = Math.log(Math.exp((t_0 + (x - Math.tan(a)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -3.5e-12:
		tmp = x + (t_0 - math.tan(a))
	elif a <= 1.25e-17:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (-1.0 + (1.0 + (math.tan(y) * math.tan(z))))))
	else:
		tmp = math.log(math.exp((t_0 + (x - math.tan(a)))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -3.5e-12)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (a <= 1.25e-17)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(-1.0 + Float64(1.0 + Float64(tan(y) * tan(z)))))));
	else
		tmp = log(exp(Float64(t_0 + Float64(x - tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -3.5e-12)
		tmp = x + (t_0 - tan(a));
	elseif (a <= 1.25e-17)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (-1.0 + (1.0 + (tan(y) * tan(z))))));
	else
		tmp = log(exp((t_0 + (x - tan(a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -3.5e-12], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-17], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-1.0 + N[(1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5e-12

   1. Initial program 79.1%

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

   if -3.5e-12 < a < 1.25e-17

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-+l- 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in a around 0 80.0%

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

      1. neg-mul-1 80.0%

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

   6. Simplified 80.0%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. 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%

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

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \left(-x\right) \]
   9. 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%

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

   10. Simplified 99.6%

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

       1. expm1-log1p-u 95.0%

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

       2. expm1-udef 95.0%

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

       3. log1p-udef 95.0%

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

       4. add-exp-log 99.7%

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

   12. Applied egg-rr 99.6%

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

   if 1.25e-17 < a

   1. Initial program 78.9%

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

      1. associate-+r- 78.9%

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

      2. add-log-exp 78.9%

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

      3. +-commutative 78.9%

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

      4. associate--l+ 79.0%

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

   3. Applied egg-rr 79.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \left(-1 + \left(1 + \tan y \cdot \tan z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\tan \left(y + z\right) + \left(x - \tan a\right)}\right)\\ \end{array} \]

Alternative 6: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -1.26 \cdot 10^{-11}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0 + \left(x - \tan a\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -1.26e-11)
     (+ x (- t_0 (tan a)))
     (if (<= a 1.25e-17)
       (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
       (log (exp (+ t_0 (- x (tan a)))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -1.26e-11) {
		tmp = x + (t_0 - tan(a));
	} else if (a <= 1.25e-17) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = log(exp((t_0 + (x - tan(a)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-1.26d-11)) then
        tmp = x + (t_0 - tan(a))
    else if (a <= 1.25d-17) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = log(exp((t_0 + (x - tan(a)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -1.26e-11) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (a <= 1.25e-17) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = Math.log(Math.exp((t_0 + (x - Math.tan(a)))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -1.26e-11:
		tmp = x + (t_0 - math.tan(a))
	elif a <= 1.25e-17:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = math.log(math.exp((t_0 + (x - math.tan(a)))))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -1.26e-11)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (a <= 1.25e-17)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = log(exp(Float64(t_0 + Float64(x - tan(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -1.26e-11)
		tmp = x + (t_0 - tan(a));
	elseif (a <= 1.25e-17)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	else
		tmp = log(exp((t_0 + (x - tan(a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -1.26e-11], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-17], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.26e-11

   1. Initial program 79.1%

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

   if -1.26e-11 < a < 1.25e-17

   1. Initial program 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-+l- 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in a around 0 80.0%

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

      1. neg-mul-1 80.0%

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

   6. Simplified 80.0%

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

      1. sub-neg 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. remove-double-neg 80.0%

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

      2. +-commutative 80.0%

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

      3. +-commutative 80.0%

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

   10. Simplified 80.0%

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

       1. +-commutative 80.0%

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

       2. +-commutative 80.0%

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

       3. tan-sum 99.6%

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

       4. div-inv 99.6%

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

       5. fma-def 99.6%

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

   12. Applied egg-rr 99.6%

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

       1. fma-udef 99.6%

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

       2. associate-*r/ 99.6%

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

       3. *-rgt-identity 99.6%

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

   14. Simplified 99.6%

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

   if 1.25e-17 < a

   1. Initial program 78.9%

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

      1. associate-+r- 78.9%

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

      2. add-log-exp 78.9%

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

      3. +-commutative 78.9%

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

      4. associate--l+ 79.0%

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

   3. Applied egg-rr 79.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\tan \left(y + z\right) + \left(x - \tan a\right)}\right)\\ \end{array} \]

Alternative 7: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.052:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 0.052) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.052) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 0.052d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.052) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 0.052:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 0.052)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 0.052)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 0.052], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0519999999999999976

   1. Initial program 86.3%

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

   2. Taylor expanded in z around 0 71.0%

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

      1. tan-quot 71.1%

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

      2. tan-quot 71.1%

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

      3. associate--l+ 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. associate-+r- 71.1%

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

      2. +-commutative 71.1%

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

      3. associate--l+ 71.1%

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

   6. Simplified 71.1%

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

   if 0.0519999999999999976 < z

   1. Initial program 60.6%

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

      1. +-commutative 60.6%

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

      2. associate-+l- 60.5%

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

   3. Applied egg-rr 60.5%

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

   4. Taylor expanded in a around 0 50.0%

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

      1. neg-mul-1 50.0%

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

   6. Simplified 50.0%

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

      1. sub-neg 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. remove-double-neg 50.0%

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

      2. +-commutative 50.0%

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

      3. +-commutative 50.0%

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

   10. Simplified 50.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.052:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 8: 79.8% 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 79.5%

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

2. Final simplification 79.5%

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

Alternative 9: 50.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. +-commutative 79.5%

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

   2. associate-+l- 79.5%

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

3. Applied egg-rr 79.5%

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

4. Taylor expanded in a around 0 51.6%

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

   1. neg-mul-1 51.6%

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

6. Simplified 51.6%

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

   1. sub-neg 51.6%

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

8. Applied egg-rr 51.6%

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

   1. remove-double-neg 51.6%

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

   2. +-commutative 51.6%

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

   3. +-commutative 51.6%

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

10. Simplified 51.6%

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

11. Final simplification 51.6%

    \[\leadsto x + \tan \left(y + z\right) \]

Alternative 10: 31.7% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

2. Taylor expanded in x around inf 29.6%

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

3. Final simplification 29.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(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))))