tan-example (used to crash)

Percentage Accurate: 79.0% → 99.6%

Time: 33.8s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% 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

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma
  (fma
   (+ (tan z) (tan y))
   (/ (/ 1.0 (fma (- (sin y)) (/ (/ (sin z) (cos z)) (cos y)) 1.0)) x)
   (/ (tan a) (- x)))
  x
  x))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return fma(fma((tan(z) + tan(y)), ((1.0 / fma(-sin(y), ((sin(z) / cos(z)) / cos(y)), 1.0)) / x), (tan(a) / -x)), x, x);
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return fma(fma(Float64(tan(z) + tan(y)), Float64(Float64(1.0 / fma(Float64(-sin(y)), Float64(Float64(sin(z) / cos(z)) / cos(y)), 1.0)) / x), Float64(tan(a) / Float64(-x))), x, x)
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[((-N[Sin[y], $MachinePrecision]) * N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma
  (fma
   (+ (tan z) (tan y))
   (/ (pow (fma (- (tan z)) (tan y) 1.0) -1.0) x)
   (/ (tan a) (- x)))
  x
  x))

C

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return fma(fma(Float64(tan(z) + tan(y)), Float64((fma(Float64(-tan(z)), tan(y), 1.0) ^ -1.0) / x), Float64(tan(a) / Float64(-x))), x, x)
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma
  (/ (- (/ (+ (tan z) (tan y)) (fma (tan y) (- (tan z)) 1.0)) (tan a)) x)
  x
  x))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 79.9%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ (- (/ (+ (tan z) (tan y)) (- 1.0 (* (tan z) (tan y)))) (tan a)) x))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = (((tan(z) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - tan(a)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - tan(a)) + x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y)))
        (t_1 (fma (fma t_0 (/ 1.0 x) (/ (tan a) (- x))) x x)))
   (if (<= a -4.9e-32)
     t_1
     (if (<= a 2.4e-9)
       (+
        (-
         (/ t_0 (fma (- (tan z)) (tan y) 1.0))
         (* (fma 0.3333333333333333 (* a a) 1.0) a))
        x)
       t_1))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = fma(fma(t_0, (1.0 / x), (tan(a) / -x)), x, x);
	double tmp;
	if (a <= -4.9e-32) {
		tmp = t_1;
	} else if (a <= 2.4e-9) {
		tmp = ((t_0 / fma(-tan(z), tan(y), 1.0)) - (fma(0.3333333333333333, (a * a), 1.0) * a)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = fma(fma(t_0, Float64(1.0 / x), Float64(tan(a) / Float64(-x))), x, x)
	tmp = 0.0
	if (a <= -4.9e-32)
		tmp = t_1;
	elseif (a <= 2.4e-9)
		tmp = Float64(Float64(Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(1.0 / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[a, -4.9e-32], t$95$1, If[LessEqual[a, 2.4e-9], N[(N[(N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8999999999999998e-32 or 2.4e-9 < a

   1. Initial program 81.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. associate-/l/ N/A

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

      5. div-sub N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 82.4%

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

   if -4.8999999999999998e-32 < a < 2.4e-9

   1. Initial program 77.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

         \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(0.3333333333333333, \color{blue}{a \cdot a}, 1\right) \cdot a\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto x + \left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \color{blue}{\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ t_1 := -\tan z\\ \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 - \tan y, \frac{-1}{\mathsf{fma}\left(\tan y, t\_1, 1\right)}, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (fma (fma (+ (tan z) (tan y)) (/ 1.0 x) (/ (tan a) (- x))) x x))
        (t_1 (- (tan z))))
   (if (<= a -4.9e-32)
     t_0
     (if (<= a 9e-11)
       (fma (- t_1 (tan y)) (/ -1.0 (fma (tan y) t_1 1.0)) (- (- x)))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = fma(fma((tan(z) + tan(y)), (1.0 / x), (tan(a) / -x)), x, x);
	double t_1 = -tan(z);
	double tmp;
	if (a <= -4.9e-32) {
		tmp = t_0;
	} else if (a <= 9e-11) {
		tmp = fma((t_1 - tan(y)), (-1.0 / fma(tan(y), t_1, 1.0)), -(-x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = fma(fma(Float64(tan(z) + tan(y)), Float64(1.0 / x), Float64(tan(a) / Float64(-x))), x, x)
	t_1 = Float64(-tan(z))
	tmp = 0.0
	if (a <= -4.9e-32)
		tmp = t_0;
	elseif (a <= 9e-11)
		tmp = fma(Float64(t_1 - tan(y)), Float64(-1.0 / fma(tan(y), t_1, 1.0)), Float64(-Float64(-x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[z], $MachinePrecision])}, If[LessEqual[a, -4.9e-32], t$95$0, If[LessEqual[a, 9e-11], N[(N[(t$95$1 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[y], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] + (-(-x))), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8999999999999998e-32 or 8.9999999999999999e-11 < a

   1. Initial program 81.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. associate-/l/ N/A

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

      5. div-sub N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 82.4%

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

   if -4.8999999999999998e-32 < a < 8.9999999999999999e-11

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.7

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

   7. Applied rewrites 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(\tan y + \tan z\right), \frac{-1}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}, -\left(-x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}, -\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan z + \tan y\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y)))
        (t_1 (fma (fma t_0 (/ 1.0 x) (/ (tan a) (- x))) x x)))
   (if (<= a -4.9e-32)
     t_1
     (if (<= a 9e-11) (- (/ t_0 (fma (tan y) (- (tan z)) 1.0)) (- x)) t_1))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double t_1 = fma(fma(t_0, (1.0 / x), (tan(a) / -x)), x, x);
	double tmp;
	if (a <= -4.9e-32) {
		tmp = t_1;
	} else if (a <= 9e-11) {
		tmp = (t_0 / fma(tan(y), -tan(z), 1.0)) - -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	t_1 = fma(fma(t_0, Float64(1.0 / x), Float64(tan(a) / Float64(-x))), x, x)
	tmp = 0.0
	if (a <= -4.9e-32)
		tmp = t_1;
	elseif (a <= 9e-11)
		tmp = Float64(Float64(t_0 / fma(tan(y), Float64(-tan(z)), 1.0)) - Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(1.0 / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[a, -4.9e-32], t$95$1, If[LessEqual[a, 9e-11], N[(N[(t$95$0 / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8999999999999998e-32 or 8.9999999999999999e-11 < a

   1. Initial program 81.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. associate-/l/ N/A

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

      5. div-sub N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 82.4%

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

   if -4.8999999999999998e-32 < a < 8.9999999999999999e-11

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.7

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

   7. Applied rewrites 77.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. *-commutative N/A

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

      12. cancel-sign-sub-inv N/A

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

      13. lift-neg.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

   9. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}} - \left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan z + \tan y}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan z + \tan y, \frac{1}{x}, \frac{\tan a}{-x}\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.6× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma (fma (+ (tan z) (tan y)) (/ 1.0 x) (/ (tan a) (- x))) x x))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 80.4%

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

11. Final simplification 80.4%

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

Alternative 9: 79.3% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ (- (/ (+ (tan z) (tan y)) 1.0) (tan a)) x))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = (((tan(z) + tan(y)) / 1.0d0) - tan(a)) + x
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return (((math.tan(z) + math.tan(y)) / 1.0) - math.tan(a)) + x

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(Float64(Float64(Float64(tan(z) + tan(y)) / 1.0) - tan(a)) + x)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (((tan(z) + tan(y)) / 1.0) - tan(a)) + x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 80.3%

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

8. Final simplification 80.3%

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

Alternative 10: 78.9% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma (- (/ (tan (+ z y)) x) (/ (tan a) x)) x x))

C

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return fma(Float64(Float64(tan(Float64(z + y)) / x) - Float64(tan(a) / x)), x, x)
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 79.9%

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

8. Final simplification 79.9%

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

Alternative 11: 78.9% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (fma (/ (- (tan (+ z y)) (tan a)) x) x x))

C

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return fma(Float64(Float64(tan(Float64(z + y)) - tan(a)) / x), x, x)
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 79.9%

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

7. Applied rewrites 79.9%

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

8. Final simplification 79.9%

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

Alternative 12: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;z + y \leq -0.04:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (+ z y) -0.04) (- (tan (+ z y)) (- x)) (+ (- (tan z) (tan a)) x)))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((z + y) <= -0.04) {
		tmp = tan((z + y)) - -x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 + y) <= (-0.04d0)) then
        tmp = tan((z + y)) - -x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((z + y) <= -0.04) {
		tmp = Math.tan((z + y)) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (z + y) <= -0.04:
		tmp = math.tan((z + y)) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (Float64(z + y) <= -0.04)
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((z + y) <= -0.04)
		tmp = tan((z + y)) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[(z + y), $MachinePrecision], -0.04], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -0.0400000000000000008

   1. Initial program 67.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.2

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

   7. Applied rewrites 46.2%

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

   if -0.0400000000000000008 < (+.f64 y z)

   1. Initial program 87.3%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.4

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

   7. Applied rewrites 70.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + y \leq -0.04:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-8}:\\ \;\;\;\;\left(\tan y - \tan a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= z 3.6e-8) (+ (- (tan y) (tan a)) x) (+ (- (tan z) (tan a)) x)))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 3.6e-8) {
		tmp = (tan(y) - tan(a)) + x;
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= 3.6d-8) then
        tmp = (tan(y) - tan(a)) + x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 3.6e-8) {
		tmp = (Math.tan(y) - Math.tan(a)) + x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if z <= 3.6e-8:
		tmp = (math.tan(y) - math.tan(a)) + x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (z <= 3.6e-8)
		tmp = Float64(Float64(tan(y) - tan(a)) + x);
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 3.6e-8)
		tmp = (tan(y) - tan(a)) + x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[z, 3.6e-8], N[(N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999981e-8

   1. Initial program 84.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 74.1

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

   9. Applied rewrites 74.1%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 74.1

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

   11. Applied rewrites 74.1%

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

   if 3.59999999999999981e-8 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 66.2

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

   5. Applied rewrites 66.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 66.2

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

   7. Applied rewrites 66.2%

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

Alternative 14: 79.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ (- (tan (+ z y)) (tan a)) x))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = (tan((z + y)) - tan(a)) + x
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(Float64(tan(Float64(z + y)) - tan(a)) + x)
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (tan((z + y)) - tan(a)) + x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Final simplification 79.9%

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

Alternative 15: 49.6% accurate, 1.9× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- (tan (+ z y)) (- x)))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = tan((z + y)) - -x
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(tan(Float64(z + y)) - Float64(-x))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = tan((z + y)) - -x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower--.f64 79.9

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

4. Applied rewrites 79.9%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 50.1

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

7. Applied rewrites 50.1%

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

Alternative 16: 31.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 = 1.0d0 / (1.0d0 / x)
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 79.8

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

4. Applied rewrites 79.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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))))