tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 31.0s

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: 79.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := \mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 0.002:\\ \;\;\;\;\left(\frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ (fma 1.0 t_0 (- (tan a))) x)))
   (if (<= (tan a) -0.04)
     t_1
     (if (<= (tan a) 0.002)
       (+
        (-
         (/ t_0 (fma (- (tan z)) (tan y) 1.0))
         (*
          (fma
           (fma 0.13333333333333333 (* a a) 0.3333333333333333)
           (* a a)
           1.0)
          a))
        x)
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = fma(1.0, t_0, -tan(a)) + x;
	double tmp;
	if (tan(a) <= -0.04) {
		tmp = t_1;
	} else if (tan(a) <= 0.002) {
		tmp = ((t_0 / fma(-tan(z), tan(y), 1.0)) - (fma(fma(0.13333333333333333, (a * a), 0.3333333333333333), (a * a), 1.0) * a)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(fma(1.0, t_0, Float64(-tan(a))) + x)
	tmp = 0.0
	if (tan(a) <= -0.04)
		tmp = t_1;
	elseif (tan(a) <= 0.002)
		tmp = Float64(Float64(Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(fma(fma(0.13333333333333333, Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.04], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 0.002], N[(N[(N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.13333333333333333 * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0400000000000000008 or 2e-3 < (tan.f64 a)

   1. Initial program 77.4%

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

      1. lift--.f64 N/A

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

      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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. clear-num N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 77.9%

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

   if -0.0400000000000000008 < (tan.f64 a) < 2e-3

   1. Initial program 76.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 0.002:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := \mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 0.002:\\ \;\;\;\;\left(\frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ (fma 1.0 t_0 (- (tan a))) x)))
   (if (<= (tan a) -0.04)
     t_1
     (if (<= (tan a) 0.002)
       (+
        (-
         (/ t_0 (fma (- (tan z)) (tan y) 1.0))
         (* (fma 0.3333333333333333 (* a a) 1.0) a))
        x)
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = fma(1.0, t_0, -tan(a)) + x;
	double tmp;
	if (tan(a) <= -0.04) {
		tmp = t_1;
	} else if (tan(a) <= 0.002) {
		tmp = ((t_0 / fma(-tan(z), tan(y), 1.0)) - (fma(0.3333333333333333, (a * a), 1.0) * a)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(fma(1.0, t_0, Float64(-tan(a))) + x)
	tmp = 0.0
	if (tan(a) <= -0.04)
		tmp = t_1;
	elseif (tan(a) <= 0.002)
		tmp = Float64(Float64(Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.04], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 0.002], N[(N[(N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0400000000000000008 or 2e-3 < (tan.f64 a)

   1. Initial program 77.4%

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

      1. lift--.f64 N/A

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

      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

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. clear-num N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 77.9%

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

   if -0.0400000000000000008 < (tan.f64 a) < 2e-3

   1. Initial program 76.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 0.002:\\ \;\;\;\;\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := \mathsf{fma}\left(1, t\_0, -\tan a\right) + x\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 0.01:\\ \;\;\;\;\frac{t\_0}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ (fma 1.0 t_0 (- (tan a))) x)))
   (if (<= (tan a) -5e-16)
     t_1
     (if (<= (tan a) 0.01)
       (- (/ t_0 (- (fma (tan z) (tan y) -1.0))) (- x))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = fma(1.0, t_0, -tan(a)) + x;
	double tmp;
	if (tan(a) <= -5e-16) {
		tmp = t_1;
	} else if (tan(a) <= 0.01) {
		tmp = (t_0 / -fma(tan(z), tan(y), -1.0)) - -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(fma(1.0, t_0, Float64(-tan(a))) + x)
	tmp = 0.0
	if (tan(a) <= -5e-16)
		tmp = t_1;
	elseif (tan(a) <= 0.01)
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(z), tan(y), -1.0))) - Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-16], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 0.01], N[(N[(t$95$0 / (-N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -5.0000000000000004e-16 or 0.0100000000000000002 < (tan.f64 a)

   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 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. clear-num N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 79.0%

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

   if -5.0000000000000004e-16 < (tan.f64 a) < 0.0100000000000000002

   1. Initial program 75.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. +-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 75.4

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

   4. Applied rewrites 75.4%

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

   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 75.3

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

   7. Applied rewrites 75.3%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(-x\right)} \]
   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. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. lower-neg.f64 97.8

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

      22. lift-fma.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-fma.f64 97.8

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

   9. Applied rewrites 97.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \mathbf{elif}\;\tan a \leq 0.01:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan z, \tan y, -1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 99.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 79.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift--.f64 N/A

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

   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

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. clear-num N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. frac-2neg N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in z around 0

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

9. Applied rewrites 77.3%

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

10. Final simplification 77.3%

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

Alternative 7: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Final simplification 76.9%

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

Alternative 8: 51.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower--.f64 76.8

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

4. Applied rewrites 76.8%

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

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 51.3

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

7. Applied rewrites 51.3%

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

8. Final simplification 51.3%

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

Alternative 9: 31.9% 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 76.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 76.8

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 29.6

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

7. Applied rewrites 29.6%

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

Reproduce

herbie shell --seed 2024277 
(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))))