tan-example (used to crash)

Percentage Accurate: 79.6% → 99.6%

Time: 41.8s

Alternatives: 12

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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

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

   3. fma-neg 99.7%

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

4. Applied egg-rr 99.7%

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

   1. log1p-expm1-u 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11} \lor \neg \left(\tan a \leq 0.04\right):\\ \;\;\;\;x + \mathsf{fma}\left(t_0, 1, -\tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (or (<= (tan a) -2e-11) (not (<= (tan a) 0.04)))
     (+ x (fma t_0 1.0 (- (tan a))))
     (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if ((tan(a) <= -2e-11) || !(tan(a) <= 0.04)) {
		tmp = x + fma(t_0, 1.0, -tan(a));
	} else {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if ((tan(a) <= -2e-11) || !(tan(a) <= 0.04))
		tmp = Float64(x + fma(t_0, 1.0, Float64(-tan(a))));
	else
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -2e-11], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 0.04]], $MachinePrecision]], N[(x + N[(t$95$0 * 1.0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -1.99999999999999988e-11 or 0.0400000000000000008 < (tan.f64 a)

   1. Initial program 77.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
   3. 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. fma-neg 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 78.1%

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

   if -1.99999999999999988e-11 < (tan.f64 a) < 0.0400000000000000008

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

      2. associate-+l- 71.0%

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

   4. Applied egg-rr 71.0%

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

   5. Taylor expanded in a around 0 71.0%

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

      1. neg-mul-1 71.0%

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

   7. Simplified 71.0%

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

      1. sub-neg 71.0%

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

      2. remove-double-neg 71.0%

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

   9. Applied egg-rr 71.0%

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

       1. tan-sum 98.2%

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

       2. div-inv 98.1%

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

   11. Applied egg-rr 98.1%

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

       1. associate-*r/ 98.2%

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

       2. *-rgt-identity 98.2%

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

   13. Simplified 98.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11} \lor \neg \left(\tan a \leq 0.04\right):\\ \;\;\;\;x + \mathsf{fma}\left(\tan y + \tan z, 1, -\tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

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

   3. fma-neg 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. tan-sum 61.1%

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

   2. div-inv 61.1%

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

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/ 61.1%

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

   2. *-rgt-identity 61.1%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;x - \frac{\sin a}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.01) (not (<= (tan a) 5e-19)))
   (- x (/ (sin a) (cos a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.01) || !(tan(a) <= 5e-19)) {
		tmp = x - (sin(a) / cos(a));
	} else {
		tmp = tan((y + z)) + (x - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-0.01d0)) .or. (.not. (tan(a) <= 5d-19))) then
        tmp = x - (sin(a) / cos(a))
    else
        tmp = tan((y + z)) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -0.01) || !(Math.tan(a) <= 5e-19)) {
		tmp = x - (Math.sin(a) / Math.cos(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.01) or not (math.tan(a) <= 5e-19):
		tmp = x - (math.sin(a) / math.cos(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.01) || !(tan(a) <= 5e-19))
		tmp = Float64(x - Float64(sin(a) / cos(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.01) || ~((tan(a) <= 5e-19)))
		tmp = x - (sin(a) / cos(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-19]], $MachinePrecision]], N[(x - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0100000000000000002 or 5.0000000000000004e-19 < (tan.f64 a)

   1. Initial program 75.1%

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

      1. +-commutative 75.1%

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

      2. associate-+l- 75.0%

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

   4. Applied egg-rr 75.0%

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

   5. Taylor expanded in y around 0 55.9%

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

   6. Taylor expanded in z around 0 36.2%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

   if -0.0100000000000000002 < (tan.f64 a) < 5.0000000000000004e-19

   1. Initial program 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-+l- 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in a around 0 73.7%

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

      1. neg-mul-1 73.7%

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

      2. unsub-neg 73.7%

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

   7. Simplified 73.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;x - \frac{\sin a}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

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

   3. fma-neg 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0 74.7%

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

6. Final simplification 74.7%

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

Alternative 7: 59.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-11} \lor \neg \left(t_0 \leq 0.002\right):\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (or (<= t_0 -5e-11) (not (<= t_0 0.002)))
     (+ x t_0)
     (+ z (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if ((t_0 <= -5e-11) || !(t_0 <= 0.002)) {
		tmp = x + t_0;
	} else {
		tmp = z + (x - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if ((t_0 <= (-5d-11)) .or. (.not. (t_0 <= 0.002d0))) then
        tmp = x + t_0
    else
        tmp = z + (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 ((t_0 <= -5e-11) || !(t_0 <= 0.002)) {
		tmp = x + t_0;
	} else {
		tmp = z + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if (t_0 <= -5e-11) or not (t_0 <= 0.002):
		tmp = x + t_0
	else:
		tmp = z + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if ((t_0 <= -5e-11) || !(t_0 <= 0.002))
		tmp = Float64(x + t_0);
	else
		tmp = Float64(z + 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 ((t_0 <= -5e-11) || ~((t_0 <= 0.002)))
		tmp = x + t_0;
	else
		tmp = z + (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[Or[LessEqual[t$95$0, -5e-11], N[Not[LessEqual[t$95$0, 0.002]], $MachinePrecision]], N[(x + t$95$0), $MachinePrecision], N[(z + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -5.00000000000000018e-11 or 2e-3 < (tan.f64 (+.f64 y z))

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. associate-+l- 68.8%

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in a around 0 44.1%

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

      1. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

      1. sub-neg 44.1%

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

      2. remove-double-neg 44.1%

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

   9. Applied egg-rr 44.1%

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

   if -5.00000000000000018e-11 < (tan.f64 (+.f64 y z)) < 2e-3

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 99.2%

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

   6. Taylor expanded in z around 0 99.2%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(y + z\right) \leq -5 \cdot 10^{-11} \lor \neg \left(\tan \left(y + z\right) \leq 0.002\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \tan \left(y + z\right)\\ \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y + z \leq 0.002:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (tan (+ y z)))))
   (if (<= (+ y z) -5e-11)
     t_0
     (if (<= (+ y z) 0.002) (+ z (- x (tan a))) (fabs t_0)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + tan((y + z));
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = t_0;
	} else if ((y + z) <= 0.002) {
		tmp = z + (x - tan(a));
	} else {
		tmp = fabs(t_0);
	}
	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 = x + tan((y + z))
    if ((y + z) <= (-5d-11)) then
        tmp = t_0
    else if ((y + z) <= 0.002d0) then
        tmp = z + (x - tan(a))
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x + Math.tan((y + z));
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = t_0;
	} else if ((y + z) <= 0.002) {
		tmp = z + (x - Math.tan(a));
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x + math.tan((y + z))
	tmp = 0
	if (y + z) <= -5e-11:
		tmp = t_0
	elif (y + z) <= 0.002:
		tmp = z + (x - math.tan(a))
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x + tan(Float64(y + z)))
	tmp = 0.0
	if (Float64(y + z) <= -5e-11)
		tmp = t_0;
	elseif (Float64(y + z) <= 0.002)
		tmp = Float64(z + Float64(x - tan(a)));
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x + tan((y + z));
	tmp = 0.0;
	if ((y + z) <= -5e-11)
		tmp = t_0;
	elseif ((y + z) <= 0.002)
		tmp = z + (x - tan(a));
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + z), $MachinePrecision], -5e-11], t$95$0, If[LessEqual[N[(y + z), $MachinePrecision], 0.002], N[(z + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -5.00000000000000018e-11

   1. Initial program 65.6%

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

      1. +-commutative 65.6%

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

      2. associate-+l- 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in a around 0 42.8%

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

      1. neg-mul-1 42.8%

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

   7. Simplified 42.8%

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

      1. sub-neg 42.8%

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

      2. remove-double-neg 42.8%

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

   9. Applied egg-rr 42.8%

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

   if -5.00000000000000018e-11 < (+.f64 y z) < 2e-3

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 99.2%

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

   6. Taylor expanded in z around 0 99.2%

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

   if 2e-3 < (+.f64 y z)

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+l- 72.9%

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

   4. Applied egg-rr 72.9%

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

   5. Taylor expanded in a around 0 45.7%

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

      1. neg-mul-1 45.7%

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

   7. Simplified 45.7%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 45.9%

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

      3. pow2 45.9%

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

      4. sub-neg 45.9%

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

      5. remove-double-neg 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. unpow2 45.9%

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

       2. rem-sqrt-square 45.9%

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

       3. +-commutative 45.9%

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

   11. Simplified 45.9%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{elif}\;y + z \leq 0.002:\\ \;\;\;\;z + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x + \tan \left(y + z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-11) (+ x (tan (+ y z))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = x + tan((y + z));
	} else {
		tmp = tan(z) + (x - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-5d-11)) then
        tmp = x + tan((y + z))
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-11) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5e-11:
		tmp = x + math.tan((y + z))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5e-11)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5e-11)
		tmp = x + tan((y + z));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5e-11], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -5.00000000000000018e-11

   1. Initial program 65.6%

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

      1. +-commutative 65.6%

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

      2. associate-+l- 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in a around 0 42.8%

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

      1. neg-mul-1 42.8%

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

   7. Simplified 42.8%

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

      1. sub-neg 42.8%

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

      2. remove-double-neg 42.8%

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

   9. Applied egg-rr 42.8%

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

   if -5.00000000000000018e-11 < (+.f64 y z)

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. associate-+l- 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in y around 0 64.2%

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

      1. tan-quot 64.1%

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

      2. expm1-log1p-u 57.6%

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

      3. expm1-udef 57.6%

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

   7. Applied egg-rr 57.6%

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

      1. expm1-def 57.6%

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

      2. expm1-log1p 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.6% 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 74.4%

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

3. Final simplification 74.4%

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

Alternative 11: 50.3% 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 74.4%

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

   1. +-commutative 74.4%

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

   2. associate-+l- 74.4%

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

4. Applied egg-rr 74.4%

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

5. Taylor expanded in a around 0 47.7%

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

   1. neg-mul-1 47.7%

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

7. Simplified 47.7%

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

   1. sub-neg 47.7%

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

   2. remove-double-neg 47.7%

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

9. Applied egg-rr 47.7%

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

10. Final simplification 47.7%

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

Alternative 12: 31.8% 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 74.4%

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

3. Taylor expanded in x around inf 30.8%

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

4. Final simplification 30.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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))))