tan-example (used to crash)

Percentage Accurate: 79.0% → 99.6%

Time: 34.0s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\tan y + \tan z\right) \cdot \mathsf{fma}\left(\tan y, \tan z, 1\right)\\ \mathsf{fma}\left({\left(\frac{t\_0}{1 - {\tan z}^{2} \cdot {\tan y}^{2}}\right)}^{2} - {\tan a}^{2}, \frac{1}{\tan a + \frac{t\_0}{1 - {\left(\tan y \cdot \tan z\right)}^{2}}}, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (+ (tan y) (tan z)) (fma (tan y) (tan z) 1.0))))
   (fma
    (-
     (pow (/ t_0 (- 1.0 (* (pow (tan z) 2.0) (pow (tan y) 2.0)))) 2.0)
     (pow (tan a) 2.0))
    (/ 1.0 (+ (tan a) (/ t_0 (- 1.0 (pow (* (tan y) (tan z)) 2.0)))))
    x)))

C

double code(double x, double y, double z, double a) {
	double t_0 = (tan(y) + tan(z)) * fma(tan(y), tan(z), 1.0);
	return fma((pow((t_0 / (1.0 - (pow(tan(z), 2.0) * pow(tan(y), 2.0)))), 2.0) - pow(tan(a), 2.0)), (1.0 / (tan(a) + (t_0 / (1.0 - pow((tan(y) * tan(z)), 2.0))))), x);
}

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(tan(y) + tan(z)) * fma(tan(y), tan(z), 1.0))
	return fma(Float64((Float64(t_0 / Float64(1.0 - Float64((tan(z) ^ 2.0) * (tan(y) ^ 2.0)))) ^ 2.0) - (tan(a) ^ 2.0)), Float64(1.0 / Float64(tan(a) + Float64(t_0 / Float64(1.0 - (Float64(tan(y) * tan(z)) ^ 2.0))))), x)
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(t$95$0 / N[(1.0 - N[(N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Tan[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Tan[a], $MachinePrecision] + N[(t$95$0 / N[(1.0 - N[Power[N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Initial program 78.4%

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

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. flip-+ N/A

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

   5. div-inv N/A

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left({\left(\frac{\left(\tan y + \tan z\right) \cdot \mathsf{fma}\left(\tan y, \tan z, 1\right)}{1 - {\tan z}^{2} \cdot {\tan y}^{2}}\right)}^{2} - {\tan a}^{2}, \frac{1}{\tan a + \frac{\left(\tan y + \tan z\right) \cdot \mathsf{fma}\left(\tan y, \tan z, 1\right)}{1 - {\left(\tan y \cdot \tan z\right)}^{2}}}, x\right) \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathsf{fma}\left({\left(\frac{t\_0 \cdot \mathsf{fma}\left(\tan y, \tan z, 1\right)}{1 - {\tan z}^{2} \cdot {\tan y}^{2}}\right)}^{2} - {\tan a}^{2}, \frac{1}{\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, \tan a\right)}, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (fma
    (-
     (pow
      (/
       (* t_0 (fma (tan y) (tan z) 1.0))
       (- 1.0 (* (pow (tan z) 2.0) (pow (tan y) 2.0))))
      2.0)
     (pow (tan a) 2.0))
    (/ 1.0 (fma (/ 1.0 (- 1.0 (* (tan y) (tan z)))) t_0 (tan a)))
    x)))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	return fma((pow(((t_0 * fma(tan(y), tan(z), 1.0)) / (1.0 - (pow(tan(z), 2.0) * pow(tan(y), 2.0)))), 2.0) - pow(tan(a), 2.0)), (1.0 / fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, tan(a))), x);
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	return fma(Float64((Float64(Float64(t_0 * fma(tan(y), tan(z), 1.0)) / Float64(1.0 - Float64((tan(z) ^ 2.0) * (tan(y) ^ 2.0)))) ^ 2.0) - (tan(a) ^ 2.0)), Float64(1.0 / fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, tan(a))), x)
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(N[(t$95$0 * N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Tan[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

   1. lift--.f64 N/A

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

   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. flip-- N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. flip-+ N/A

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

   5. div-inv N/A

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

6. Applied rewrites 99.6%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. remove-double-neg N/A

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

   9. lower-fma.f64 N/A

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

10. Applied rewrites 99.5%

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

Reproduce

herbie shell --seed 2024230 
(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))))