tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 32.9s

Alternatives: 18

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. frac-sub N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. lower-fma.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. tan-sum N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate--r+ N/A

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

   5. neg-sub0 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lower--.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-lft-neg-out N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 99.6

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. lift-*.f64 N/A

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

   13. metadata-eval N/A

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

8. Applied rewrites 99.6%

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

Alternative 5: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ \mathbf{if}\;a \leq -0.0125:\\ \;\;\;\;\mathsf{fma}\left(-\sin \left(z + y\right), \frac{-1}{\cos \left(z + y\right)}, x - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0125:\\ \;\;\;\;\mathsf{fma}\left(t\_0 - \tan y, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, x - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))))
   (if (<= a -0.0125)
     (fma (- (sin (+ z y))) (/ -1.0 (cos (+ z y))) (- x (tan a)))
     (if (<= a 0.0125)
       (fma
        (- t_0 (tan y))
        (/ -1.0 (fma t_0 (tan y) 1.0))
        (- x (* (fma 0.3333333333333333 (* a a) 1.0) a)))
       (- x (+ (/ (+ (tan z) (tan y)) -1.0) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double tmp;
	if (a <= -0.0125) {
		tmp = fma(-sin((z + y)), (-1.0 / cos((z + y))), (x - tan(a)));
	} else if (a <= 0.0125) {
		tmp = fma((t_0 - tan(y)), (-1.0 / fma(t_0, tan(y), 1.0)), (x - (fma(0.3333333333333333, (a * a), 1.0) * a)));
	} else {
		tmp = x - (((tan(z) + tan(y)) / -1.0) + tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	tmp = 0.0
	if (a <= -0.0125)
		tmp = fma(Float64(-sin(Float64(z + y))), Float64(-1.0 / cos(Float64(z + y))), Float64(x - tan(a)));
	elseif (a <= 0.0125)
		tmp = fma(Float64(t_0 - tan(y)), Float64(-1.0 / fma(t_0, tan(y), 1.0)), Float64(x - Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a)));
	else
		tmp = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / -1.0) + tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, If[LessEqual[a, -0.0125], N[((-N[Sin[N[(z + y), $MachinePrecision]], $MachinePrecision]) * N[(-1.0 / N[Cos[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0125], N[(N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.012500000000000001

   1. Initial program 74.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. frac-2neg N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 74.8

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

   4. Applied rewrites 74.8%

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

   if -0.012500000000000001 < a < 0.012500000000000001

   1. Initial program 77.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 0.012500000000000001 < a

   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-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. tan-sum N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate--r+ N/A

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

      5. neg-sub0 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 99.7

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 84.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0125:\\ \;\;\;\;\mathsf{fma}\left(-\sin \left(z + y\right), \frac{-1}{\cos \left(z + y\right)}, x - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0125:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, x - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1} + \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;\tan a \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 0.07:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= (tan a) -0.15)
     t_0
     (if (<= (tan a) 0.07) (- (tan (+ z y)) (- x)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (tan(a) <= -0.15) {
		tmp = t_0;
	} else if (tan(a) <= 0.07) {
		tmp = tan((z + y)) - -x;
	} else {
		tmp = 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(a)
    if (tan(a) <= (-0.15d0)) then
        tmp = t_0
    else if (tan(a) <= 0.07d0) then
        tmp = tan((z + y)) - -x
    else
        tmp = 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(a);
	double tmp;
	if (Math.tan(a) <= -0.15) {
		tmp = t_0;
	} else if (Math.tan(a) <= 0.07) {
		tmp = Math.tan((z + y)) - -x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if math.tan(a) <= -0.15:
		tmp = t_0
	elif math.tan(a) <= 0.07:
		tmp = math.tan((z + y)) - -x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (tan(a) <= -0.15)
		tmp = t_0;
	elseif (tan(a) <= 0.07)
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (tan(a) <= -0.15)
		tmp = t_0;
	elseif (tan(a) <= 0.07)
		tmp = tan((z + y)) - -x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.15], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 0.07], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.149999999999999994 or 0.070000000000000007 < (tan.f64 a)

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.2%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.2%

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

   if -0.149999999999999994 < (tan.f64 a) < 0.070000000000000007

   1. Initial program 78.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.2

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

   4. Applied rewrites 78.2%

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

   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 71.7

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

   7. Applied rewrites 71.7%

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

Alternative 7: 80.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x - (((tan(z) + tan(y)) / -1.0) + 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(z) + tan(y)) / (-1.0d0)) + tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate--r+ N/A

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

   5. neg-sub0 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lower--.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 99.6

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. lift-*.f64 N/A

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

   13. metadata-eval N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 78.7%

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

12. Final simplification 78.7%

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

Alternative 8: 80.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 78.7%

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

8. Final simplification 78.7%

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

Alternative 9: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -8e-13) (- (tan (+ z y)) (- x)) (- x (- (tan a) (tan z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -8e-13) {
		tmp = tan((z + y)) - -x;
	} else {
		tmp = x - (tan(a) - tan(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-8d-13)) then
        tmp = tan((z + y)) - -x
    else
        tmp = x - (tan(a) - tan(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -8e-13) {
		tmp = Math.tan((z + y)) - -x;
	} else {
		tmp = x - (Math.tan(a) - Math.tan(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if y <= -8e-13:
		tmp = math.tan((z + y)) - -x
	else:
		tmp = x - (math.tan(a) - math.tan(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (y <= -8e-13)
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	else
		tmp = Float64(x - Float64(tan(a) - tan(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (y <= -8e-13)
		tmp = tan((z + y)) - -x;
	else
		tmp = x - (tan(a) - tan(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[y, -8e-13], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(x - N[(N[Tan[a], $MachinePrecision] - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.0000000000000002e-13

   1. Initial program 60.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 60.0

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

   4. Applied rewrites 60.0%

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

   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 36.8

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

   7. Applied rewrites 36.8%

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

   if -8.0000000000000002e-13 < y

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 73.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\tan a - \tan z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 78.6%

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

3. Final simplification 78.6%

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

Alternative 11: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.62:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.75:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.62)
     t_0
     (if (<= a 0.75)
       (-
        (tan (+ z y))
        (-
         (*
          (fma
           (fma
            (fma 0.05396825396825397 (* a a) 0.13333333333333333)
            (* a a)
            0.3333333333333333)
           (* a a)
           1.0)
          a)
         x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.62) {
		tmp = t_0;
	} else if (a <= 0.75) {
		tmp = tan((z + y)) - ((fma(fma(fma(0.05396825396825397, (a * a), 0.13333333333333333), (a * a), 0.3333333333333333), (a * a), 1.0) * a) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.62)
		tmp = t_0;
	elseif (a <= 0.75)
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(fma(fma(0.05396825396825397, Float64(a * a), 0.13333333333333333), Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a) - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.62], t$95$0, If[LessEqual[a, 0.75], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(N[(0.05396825396825397 * N[(a * a), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.619999999999999996 or 0.75 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.6%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

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

   if -0.619999999999999996 < a < 0.75

   1. Initial program 78.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)} - x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a} - x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a} - x\right) \]

      3. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) + 1\right)} \cdot a - x\right) \]

      4. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\left(\color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot {a}^{2}} + 1\right) \cdot a - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right), {a}^{2}, 1\right)} \cdot a - x\right) \]

      6. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right) + \frac{1}{3}}, {a}^{2}, 1\right) \cdot a - x\right) \]

      7. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right) \cdot {a}^{2}} + \frac{1}{3}, {a}^{2}, 1\right) \cdot a - x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}, {a}^{2}, \frac{1}{3}\right)}, {a}^{2}, 1\right) \cdot a - x\right) \]

      9. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{17}{315} \cdot {a}^{2} + \frac{2}{15}}, {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{17}{315}, {a}^{2}, \frac{2}{15}\right)}, {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      11. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, \color{blue}{a \cdot a}, \frac{2}{15}\right), {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, \color{blue}{a \cdot a}, \frac{2}{15}\right), {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      13. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, a \cdot a, \frac{2}{15}\right), \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, a \cdot a, \frac{2}{15}\right), \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      15. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{315}, a \cdot a, \frac{2}{15}\right), a \cdot a, \frac{1}{3}\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

      16. lower-*.f64 78.0

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a} - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.31:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.72:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.31)
     t_0
     (if (<= a 0.72)
       (-
        (tan (+ z y))
        (-
         (*
          (fma
           (fma 0.13333333333333333 (* a a) 0.3333333333333333)
           (* a a)
           1.0)
          a)
         x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.31) {
		tmp = t_0;
	} else if (a <= 0.72) {
		tmp = tan((z + y)) - ((fma(fma(0.13333333333333333, (a * a), 0.3333333333333333), (a * a), 1.0) * a) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.31)
		tmp = t_0;
	elseif (a <= 0.72)
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(fma(0.13333333333333333, Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a) - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.31], t$95$0, If[LessEqual[a, 0.72], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(0.13333333333333333 * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.309999999999999998 or 0.71999999999999997 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.6%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

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

   if -0.309999999999999998 < a < 0.71999999999999997

   1. Initial program 78.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)} - x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) \cdot a} - x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) \cdot a} - x\right) \]

      3. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) + 1\right)} \cdot a - x\right) \]

      4. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\left(\color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot {a}^{2}} + 1\right) \cdot a - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}, {a}^{2}, 1\right)} \cdot a - x\right) \]

      6. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {a}^{2} + \frac{1}{3}}, {a}^{2}, 1\right) \cdot a - x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{15}, {a}^{2}, \frac{1}{3}\right)}, {a}^{2}, 1\right) \cdot a - x\right) \]

      8. unpow2 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      10. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, a \cdot a, \frac{1}{3}\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

      11. lower-*.f64 77.9

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a} - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.07:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.076:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.07)
     t_0
     (if (<= a 0.076)
       (- (tan (+ z y)) (- (* (fma 0.3333333333333333 (* a a) 1.0) a) x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.07) {
		tmp = t_0;
	} else if (a <= 0.076) {
		tmp = tan((z + y)) - ((fma(0.3333333333333333, (a * a), 1.0) * a) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.07)
		tmp = t_0;
	elseif (a <= 0.076)
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a) - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.07], t$95$0, If[LessEqual[a, 0.076], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.070000000000000007 or 0.0759999999999999981 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.6%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

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

   if -0.070000000000000007 < a < 0.0759999999999999981

   1. Initial program 78.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.8

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

   7. Applied rewrites 77.8%

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

Alternative 14: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.031:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.04:\\ \;\;\;\;\tan \left(z + y\right) - \left(a - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.031) t_0 (if (<= a 0.04) (- (tan (+ z y)) (- a x)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.031) {
		tmp = t_0;
	} else if (a <= 0.04) {
		tmp = tan((z + y)) - (a - x);
	} else {
		tmp = 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(a)
    if (a <= (-0.031d0)) then
        tmp = t_0
    else if (a <= 0.04d0) then
        tmp = tan((z + y)) - (a - x)
    else
        tmp = 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(a);
	double tmp;
	if (a <= -0.031) {
		tmp = t_0;
	} else if (a <= 0.04) {
		tmp = Math.tan((z + y)) - (a - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.031:
		tmp = t_0
	elif a <= 0.04:
		tmp = math.tan((z + y)) - (a - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.031)
		tmp = t_0;
	elseif (a <= 0.04)
		tmp = Float64(tan(Float64(z + y)) - Float64(a - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.031)
		tmp = t_0;
	elseif (a <= 0.04)
		tmp = tan((z + y)) - (a - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.031], t$95$0, If[LessEqual[a, 0.04], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.031 or 0.0400000000000000008 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.6%

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

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

   if -0.031 < a < 0.0400000000000000008

   1. Initial program 78.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 77.6

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

   7. Applied rewrites 77.6%

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

Alternative 15: 42.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \tan a \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 39.9%

   \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]
9. Step-by-step derivation

10. Applied rewrites 39.9%

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

Alternative 16: 31.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / x)
end function

Java

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

Python

def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

function code(x, y, z, a)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.4

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 30.7

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

7. Applied rewrites 30.7%

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

Alternative 17: 22.7% accurate, 52.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 39.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 22.0%

    \[\leadsto x - a \]
12. Add Preprocessing

Alternative 18: 3.5% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := (-a)

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 39.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 22.0%

    \[\leadsto x - a \]

12. Taylor expanded in a around inf

    \[\leadsto -1 \cdot a \]
13. Step-by-step derivation

14. Applied rewrites 3.3%

    \[\leadsto -a \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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))))