tan-example (used to crash)

Percentage Accurate: 80.5% → 99.7%

Time: 29.5s

Alternatives: 9

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / fma(Float64(-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] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

Alternative 2: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ t_1 := t\_0 - \tan y\\ \mathbf{if}\;a \leq -0.0285 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.13333333333333333, -0.3333333333333333\right), a \cdot a, -1\right), a, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))) (t_1 (- t_0 (tan y))))
   (if (or (<= a -0.0285) (not (<= a 230000000000.0)))
     (fma t_1 (pow -1.0 -1.0) (- x (tan a)))
     (fma
      t_1
      (/ -1.0 (fma t_0 (tan y) 1.0))
      (fma
       (fma
        (fma (* a a) -0.13333333333333333 -0.3333333333333333)
        (* a a)
        -1.0)
       a
       x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double t_1 = t_0 - tan(y);
	double tmp;
	if ((a <= -0.0285) || !(a <= 230000000000.0)) {
		tmp = fma(t_1, pow(-1.0, -1.0), (x - tan(a)));
	} else {
		tmp = fma(t_1, (-1.0 / fma(t_0, tan(y), 1.0)), fma(fma(fma((a * a), -0.13333333333333333, -0.3333333333333333), (a * a), -1.0), a, x));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	t_1 = Float64(t_0 - tan(y))
	tmp = 0.0
	if ((a <= -0.0285) || !(a <= 230000000000.0))
		tmp = fma(t_1, (-1.0 ^ -1.0), Float64(x - tan(a)));
	else
		tmp = fma(t_1, Float64(-1.0 / fma(t_0, tan(y), 1.0)), fma(fma(fma(Float64(a * a), -0.13333333333333333, -0.3333333333333333), Float64(a * a), -1.0), a, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, Block[{t$95$1 = N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -0.0285], N[Not[LessEqual[a, 230000000000.0]], $MachinePrecision]], N[(t$95$1 * N[Power[-1.0, -1.0], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * -0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.028500000000000001 or 2.3e11 < a

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 82.2%

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

   if -0.028500000000000001 < a < 2.3e11

   1. Initial program 82.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0285 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.13333333333333333, -0.3333333333333333\right), a \cdot a, -1\right), a, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ t_1 := t\_0 - \tan y\\ \mathbf{if}\;a \leq -0.026 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))) (t_1 (- t_0 (tan y))))
   (if (or (<= a -0.026) (not (<= a 230000000000.0)))
     (fma t_1 (pow -1.0 -1.0) (- x (tan a)))
     (fma
      t_1
      (/ -1.0 (fma t_0 (tan y) 1.0))
      (fma (fma (* a a) -0.3333333333333333 -1.0) a x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double t_1 = t_0 - tan(y);
	double tmp;
	if ((a <= -0.026) || !(a <= 230000000000.0)) {
		tmp = fma(t_1, pow(-1.0, -1.0), (x - tan(a)));
	} else {
		tmp = fma(t_1, (-1.0 / fma(t_0, tan(y), 1.0)), fma(fma((a * a), -0.3333333333333333, -1.0), a, x));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	t_1 = Float64(t_0 - tan(y))
	tmp = 0.0
	if ((a <= -0.026) || !(a <= 230000000000.0))
		tmp = fma(t_1, (-1.0 ^ -1.0), Float64(x - tan(a)));
	else
		tmp = fma(t_1, Float64(-1.0 / fma(t_0, tan(y), 1.0)), fma(fma(Float64(a * a), -0.3333333333333333, -1.0), a, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, Block[{t$95$1 = N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -0.026], N[Not[LessEqual[a, 230000000000.0]], $MachinePrecision]], N[(t$95$1 * N[Power[-1.0, -1.0], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0259999999999999988 or 2.3e11 < a

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 82.2%

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

   if -0.0259999999999999988 < a < 2.3e11

   1. Initial program 82.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.026 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan z\\ t_1 := t\_0 - \tan y\\ \mathbf{if}\;a \leq -0.0135 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-1}{\mathsf{fma}\left(t\_0, \tan y, 1\right)}, x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (tan z))) (t_1 (- t_0 (tan y))))
   (if (or (<= a -0.0135) (not (<= a 230000000000.0)))
     (fma t_1 (pow -1.0 -1.0) (- x (tan a)))
     (fma t_1 (/ -1.0 (fma t_0 (tan y) 1.0)) (- x a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z);
	double t_1 = t_0 - tan(y);
	double tmp;
	if ((a <= -0.0135) || !(a <= 230000000000.0)) {
		tmp = fma(t_1, pow(-1.0, -1.0), (x - tan(a)));
	} else {
		tmp = fma(t_1, (-1.0 / fma(t_0, tan(y), 1.0)), (x - a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(-tan(z))
	t_1 = Float64(t_0 - tan(y))
	tmp = 0.0
	if ((a <= -0.0135) || !(a <= 230000000000.0))
		tmp = fma(t_1, (-1.0 ^ -1.0), Float64(x - tan(a)));
	else
		tmp = fma(t_1, Float64(-1.0 / fma(t_0, tan(y), 1.0)), Float64(x - a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = (-N[Tan[z], $MachinePrecision])}, Block[{t$95$1 = N[(t$95$0 - N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -0.0135], N[Not[LessEqual[a, 230000000000.0]], $MachinePrecision]], N[(t$95$1 * N[Power[-1.0, -1.0], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(t$95$0 * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0134999999999999998 or 2.3e11 < a

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 82.2%

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

   if -0.0134999999999999998 < a < 2.3e11

   1. Initial program 82.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0135 \lor \neg \left(a \leq 230000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, \frac{-1}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}, x - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\tan z\right) - \tan y\\ \mathbf{if}\;a \leq -2.15 \cdot 10^{-14} \lor \neg \left(a \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_0, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- (- (tan z)) (tan y))))
   (if (or (<= a -2.15e-14) (not (<= a 5.2e-14)))
     (fma t_0 (pow -1.0 -1.0) (- x (tan a)))
     (- (/ t_0 (fma (tan z) (tan y) -1.0)) (- x)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = -tan(z) - tan(y);
	double tmp;
	if ((a <= -2.15e-14) || !(a <= 5.2e-14)) {
		tmp = fma(t_0, pow(-1.0, -1.0), (x - tan(a)));
	} else {
		tmp = (t_0 / fma(tan(z), tan(y), -1.0)) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(-tan(z)) - tan(y))
	tmp = 0.0
	if ((a <= -2.15e-14) || !(a <= 5.2e-14))
		tmp = fma(t_0, (-1.0 ^ -1.0), Float64(x - tan(a)));
	else
		tmp = Float64(Float64(t_0 / fma(tan(z), tan(y), -1.0)) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[((-N[Tan[z], $MachinePrecision]) - N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -2.15e-14], N[Not[LessEqual[a, 5.2e-14]], $MachinePrecision]], N[(t$95$0 * N[Power[-1.0, -1.0], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.14999999999999999e-14 or 5.19999999999999993e-14 < a

   1. Initial program 80.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 81.6%

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

   if -2.14999999999999999e-14 < a < 5.19999999999999993e-14

   1. Initial program 83.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
   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 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. *-commutative N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. remove-double-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lift-neg.f64 N/A

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

      19. remove-double-neg N/A

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

      20. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-14} \lor \neg \left(a \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\tan z\right) - \tan y, {-1}^{-1}, x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan z\right) - \tan y}{\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. 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. Taylor expanded in y around 0

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

7. Applied rewrites 82.4%

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

8. Final simplification 82.4%

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

Alternative 7: 80.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.0

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 82.0%

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

9. Applied rewrites 82.1%

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

Alternative 8: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

Alternative 9: 51.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 82.0

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \tan \left(z + y\right) - \color{blue}{-1 \cdot x} \]
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 53.4

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

7. Applied rewrites 53.4%

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

Reproduce

herbie shell --seed 2024322 
(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))))