tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 28.6s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (tan y) (tan z)) (- 1.0 (+ 1.0 (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)) / (1.0 - (1.0 + 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(1.0 - Float64(1.0 + 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[(1.0 - N[(1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

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

3. Applied egg-rr 99.7%

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

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 91.9%

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

   2. expm1-udef 91.9%

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

   3. log1p-udef 91.9%

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

   4. add-exp-log 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   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%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;\left(x + \frac{1}{\cos \left(y + z\right) \cdot \frac{1}{\sin \left(y + z\right)}}\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.04)
   (- (+ x (/ 1.0 (* (cos (+ y z)) (/ 1.0 (sin (+ y z)))))) (tan a))
   (if (<= (tan a) 5e-6)
     (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- x a))
     (+ x (- (tan (+ y z)) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.04) {
		tmp = (x + (1.0 / (cos((y + z)) * (1.0 / sin((y + z)))))) - tan(a);
	} else if (tan(a) <= 5e-6) {
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	} else {
		tmp = x + (tan((y + 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 (tan(a) <= (-0.04d0)) then
        tmp = (x + (1.0d0 / (cos((y + z)) * (1.0d0 / sin((y + z)))))) - tan(a)
    else if (tan(a) <= 5d-6) then
        tmp = ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) + (x - a)
    else
        tmp = x + (tan((y + 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 (Math.tan(a) <= -0.04) {
		tmp = (x + (1.0 / (Math.cos((y + z)) * (1.0 / Math.sin((y + z)))))) - Math.tan(a);
	} else if (Math.tan(a) <= 5e-6) {
		tmp = ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	} else {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.04:
		tmp = (x + (1.0 / (math.cos((y + z)) * (1.0 / math.sin((y + z)))))) - math.tan(a)
	elif math.tan(a) <= 5e-6:
		tmp = ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	else:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.04)
		tmp = Float64(Float64(x + Float64(1.0 / Float64(cos(Float64(y + z)) * Float64(1.0 / sin(Float64(y + z)))))) - tan(a));
	elseif (tan(a) <= 5e-6)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.04)
		tmp = (x + (1.0 / (cos((y + z)) * (1.0 / sin((y + z)))))) - tan(a);
	elseif (tan(a) <= 5e-6)
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	else
		tmp = x + (tan((y + z)) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.04], N[(N[(x + N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-6], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0400000000000000008

   1. Initial program 86.7%

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

      1. associate-+r- 86.7%

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

   3. Simplified 86.7%

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

      1. tan-quot 86.7%

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

      2. clear-num 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. div-inv 86.8%

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

      2. +-commutative 86.8%

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

      3. +-commutative 86.8%

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

   7. Applied egg-rr 86.8%

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

   if -0.0400000000000000008 < (tan.f64 a) < 5.00000000000000041e-6

   1. Initial program 77.6%

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

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

   3. Applied egg-rr 99.7%

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

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

   5. Simplified 99.7%

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

      1. tan-sum 77.6%

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

      2. add-sqr-sqrt 45.4%

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

      3. sqrt-unprod 61.3%

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

      4. pow2 61.3%

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

   7. Applied egg-rr 61.3%

      \[\leadsto x + \left(\color{blue}{\sqrt{{\tan \left(y + z\right)}^{2}}} - \tan a\right) \]
   8. Step-by-step derivation

      1. unpow2 61.3%

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

      2. rem-sqrt-square 61.3%

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

      3. +-commutative 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in a around 0 61.3%

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

       1. rem-square-sqrt 45.4%

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

       2. fabs-sqr 45.4%

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

       3. rem-square-sqrt 77.6%

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

       4. +-commutative 77.6%

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

       5. +-commutative 77.6%

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

       6. mul-1-neg 77.6%

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

       7. unsub-neg 77.6%

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

   12. Simplified 77.6%

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

       1. tan-sum 99.7%

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

       2. +-commutative 99.7%

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

   14. Applied egg-rr 99.7%

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

   if 5.00000000000000041e-6 < (tan.f64 a)

   1. Initial program 75.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;\left(x + \frac{1}{\cos \left(y + z\right) \cdot \frac{1}{\sin \left(y + z\right)}}\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternative 3: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left(x + \frac{1}{\cos \left(y + z\right) \cdot \frac{1}{\sin \left(y + z\right)}}\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 0.1:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -1e-5)
   (- (+ x (/ 1.0 (* (cos (+ y z)) (/ 1.0 (sin (+ y z)))))) (tan a))
   (if (<= (tan a) 0.1)
     (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
     (+ x (- (tan (+ y z)) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -1e-5) {
		tmp = (x + (1.0 / (cos((y + z)) * (1.0 / sin((y + z)))))) - tan(a);
	} else if (tan(a) <= 0.1) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + (tan((y + 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 (tan(a) <= (-1d-5)) then
        tmp = (x + (1.0d0 / (cos((y + z)) * (1.0d0 / sin((y + z)))))) - tan(a)
    else if (tan(a) <= 0.1d0) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + (tan((y + 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 (Math.tan(a) <= -1e-5) {
		tmp = (x + (1.0 / (Math.cos((y + z)) * (1.0 / Math.sin((y + z)))))) - Math.tan(a);
	} else if (Math.tan(a) <= 0.1) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -1e-5:
		tmp = (x + (1.0 / (math.cos((y + z)) * (1.0 / math.sin((y + z)))))) - math.tan(a)
	elif math.tan(a) <= 0.1:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -1e-5)
		tmp = Float64(Float64(x + Float64(1.0 / Float64(cos(Float64(y + z)) * Float64(1.0 / sin(Float64(y + z)))))) - tan(a));
	elseif (tan(a) <= 0.1)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -1e-5)
		tmp = (x + (1.0 / (cos((y + z)) * (1.0 / sin((y + z)))))) - tan(a);
	elseif (tan(a) <= 0.1)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + (tan((y + z)) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -1e-5], N[(N[(x + N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.1], 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[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.9%

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

      1. associate-+r- 86.9%

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

   3. Simplified 86.9%

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

      1. tan-quot 86.8%

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

      2. clear-num 86.9%

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

   5. Applied egg-rr 86.9%

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

      1. div-inv 86.9%

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

      2. +-commutative 86.9%

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

      3. +-commutative 86.9%

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

   7. Applied egg-rr 86.9%

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

   if -1.00000000000000008e-5 < (tan.f64 a) < 0.10000000000000001

   1. Initial program 77.0%

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

   2. Taylor expanded in a around 0 76.9%

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

      1. *-rgt-identity 76.9%

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

      2. *-commutative 76.9%

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

      3. associate-*l/ 76.8%

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

      4. *-commutative 76.8%

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

      5. associate-*r/ 76.9%

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

      6. *-rgt-identity 76.9%

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

      7. +-commutative 76.9%

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

      8. +-commutative 76.9%

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

   4. Simplified 76.9%

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

      1. quot-tan 76.9%

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

      2. +-commutative 76.9%

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

      3. tan-sum 98.3%

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

      4. div-inv 98.3%

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

      5. fma-def 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. fma-udef 98.3%

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

      2. associate-*r/ 98.3%

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

      3. *-rgt-identity 98.3%

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

   8. Simplified 98.3%

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

   if 0.10000000000000001 < (tan.f64 a)

   1. Initial program 76.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left(x + \frac{1}{\cos \left(y + z\right) \cdot \frac{1}{\sin \left(y + z\right)}}\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 0.1:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternative 4: 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 79.4%

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

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

3. Applied egg-rr 99.7%

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

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 91.9%

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

   2. expm1-udef 91.9%

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

   3. log1p-udef 91.9%

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

   4. add-exp-log 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   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%

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

9. Simplified 99.7%

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

    1. sub-neg 99.7%

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

    2. associate--r+ 99.7%

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

    3. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

    1. sub-neg 99.7%

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

    2. sub0-neg 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 5: 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 79.4%

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

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

3. Applied egg-rr 99.7%

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

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

2. Final simplification 79.4%

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

Alternative 7: 51.0% 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 79.4%

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

2. Taylor expanded in a around 0 51.3%

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

   1. *-rgt-identity 51.3%

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

   2. *-commutative 51.3%

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

   3. associate-*l/ 51.3%

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

   4. *-commutative 51.3%

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

   5. associate-*r/ 51.3%

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

   6. *-rgt-identity 51.3%

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

   7. +-commutative 51.3%

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

   8. +-commutative 51.3%

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

4. Simplified 51.3%

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

   1. quot-tan 51.3%

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

   2. +-commutative 51.3%

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

   3. expm1-log1p-u 50.6%

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

   4. expm1-udef 50.6%

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

   5. +-commutative 50.6%

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

6. Applied egg-rr 50.6%

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

   1. expm1-def 50.6%

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

   2. expm1-log1p 51.3%

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

   3. +-commutative 51.3%

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

8. Simplified 51.3%

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

9. Final simplification 51.3%

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

Alternative 8: 32.2% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

2. Taylor expanded in x around inf 33.2%

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

3. Final simplification 33.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023230 
(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))))