tan-example (used to crash)

Percentage Accurate: 79.9% → 99.7%

Time: 30.1s

Alternatives: 13

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: 79.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, x - \mathsf{fma}\left(a, 0.3333333333333333 \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.02)
     t_1
     (if (<= (tan a) 5e-46)
       (fma
        (/ 1.0 (- 1.0 (* (tan y) (tan z))))
        t_0
        (- x (fma a (* 0.3333333333333333 (* a a)) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_1;
	} else if (tan(a) <= 5e-46) {
		tmp = fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, (x - fma(a, (0.3333333333333333 * (a * a)), a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_1;
	elseif (tan(a) <= 5e-46)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, Float64(x - fma(a, Float64(0.3333333333333333 * Float64(a * a)), a)));
	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[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-46], N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - N[(a * N[(0.3333333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 4.99999999999999992e-46 < (tan.f64 a)

   1. Initial program 80.3%

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

      1. lift--.f64 N/A

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

      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. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-tan.f64 N/A

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

      17. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 80.7%

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

   if -0.0200000000000000004 < (tan.f64 a) < 4.99999999999999992e-46

   1. Initial program 79.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

   7. Applied rewrites 99.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, x - \mathsf{fma}\left(a, 0.3333333333333333 \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))))